o
o
o 00
Sman airplane
MngJev 1I".un
Satell ite
Mars prober
Cell pnncture
Automobile
Optical III icroscope
Imegrated USM
Camera
For semiconductor manufacture
Mobi le phone
Walch
Color copier
Micro robot
piezoelectric act uator
Morphing \\;ng(by PZT actuator)
For aSlronom icaltelescope
Color figure 1
Aerial robot
Ultrasonic motors in applications
Space manipulator
Non-magnetic USM
V shape linear US
Ponable gasoline generator
USM with encoder
Non-contact USM
BUllerfty s hape linear USM
Surve il lance camera platfonn
Color figure 2
Three DOr USM
x- Y slage driven by rotarY USMs
Vacuum cleaning robot
Bar-type USM
Mode conversion type USMs
x-Y stage driven by line..."Ir
Joilll robot
Some ultrasonic motors developed by PDLab at NUAA and their applications
Active Il utter suppression system
MRI syringe
Fig. 1.1 I ( b )
Fig, 1.1 I ( a )
Fig,I,6
Fig. I,25
Fig. 1.1 9
Fig.1.8(b)
Fig. l.ll (d)
Fig. I.II (c)
Fig, 1.28
Fig. 1.26
Fig.2.15
- I
).03 0 .02 00 1
»tn
Fig.5.2
0
~.OI"(Wl--OOJ -(J.O} Fig.5 , 19 ( b )
Fig,5. 19(a )
Color figure 3
Some colored figures in the book
Fig.6. 15(a)
Fig,6 ,15(b)
Fig.6.19
Fig.I I.25(b)
Fig. IO.2?
Fig.6.22
Fig.6.20
Fig. l l.29
Fig. 11 .20
Fig.14.1
Fig.12.35
Fig.11.24(b)
Fig.14 .6
e Fig. 14.?
Fig. 14. 17(b)
Fig. 14. 14
Color figure 4
Some colored figures in the book
Fig. 14.20
Fig. 15.32
Chunsheng Zhao
Ultrasonic Motors Technologies and Applications
Chunsheng Zhao
Ultrasonic Motors Technologies and Applications
Wi th 564 figures, 14 of them in eolor
e;e Science Press Beijing .dl
'.£l Springer
Author
Chunsheng Zhao Precision Driving La bora tory Nanjing University of Aeronautics and Astronautics Nanjing 210016, China Email:
[email protected]
ISBN 978-7-03-029018-9 Science Press, Beijing Springer ISBN 978-3-642-15304-4 e-ISBN 978-3-642-15305-1 Springer Heidelberg Dordrech t London New Y or k Library of Congress Control Number: 2010932502
© Science Press Beijing and Springer- Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights arc reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The usc of general descriptive names, registered names, trademarks, etc. in this publication docs not imply, even in the absence of a specific statement, that such names arc exempt from the relevant protective laws and regulations and therefore free for general use.
Cover design: Frido Sleinen-Broo, EStudio Calamar, Spain Printed on acid- free paper Springer is part of Springer Science+Business MediaC www.springer.com)
Foreword by Jorg Wallaschek Piezoelectric Ultrasonic Motors are fascinating actuators. They combine fast dynamics and high force, can be adopted to a wide range of applications, and offer many advantages in comparison to electromagnetic and other motors. As a consequence, research in the field of piezoelectric ultrasonic motors has attracted scientists from all over the world, and many valuable contributions have been made during the past decades, while some open research questions still wait for an answer. Professor Chunsheng Zhao is a pioneer in the research of piezoelectric ultrasonic motors. His contributions to the scientific progress in this field inelude theoretical and experimental works, whose results arc documented in a large number of scientific papers and patents, and he has also developed many different prototypes of piezoelectric ultrasonic motors. Professor Chunsheng Zhao was the organizer of important international conferences and he also was the founder and director of a very successful research lab which has been recognized world wide. The present book summarizes not only the results of Prof. Zhao's work, but also provides an excellent survey on the state of the art of piezoelectric ultrasonic motors, which can be used as a textbook for teaching as well as a reference for experts in the field. I sincerely wish that it will be of practical usc to a hopefully very large number of readers and that it will also stimulate further research in the field. January 26th, 2010 Prof. Dr. - lng. habil. J brg Wallaschek
Leibniz Universitiit Hannover
Foreword by Bangchun Wen Vibratory utilization engineering is a new branch of vibration science which has been developing since the second half of the 20th century. Due to the academic significance and important applications of the field, the Vibratory Utilization Engineering Specialty Commission (VUESC) was founded in affiliation with the Chinese Association of Vibration Engineering in the last few years. It is aimed at promoting wider use and further development of the discipline through regular conferences and communication. Prof. Chunsheng Zhao, a well-known expert in vibration engineering, has researched vibration and vibration utilization engineering for more than 40 years, and has achieved fruitful results in both theory and engineering applications of vibration. For the past 15 years, Prof. Zhao has specialized in ultrasonic motors. He and his research team have developed more than 30 new types of the ultrasonic motors and corresponding drivers with proprietary intellectual rights, 71 invention patents awarded and pending in China and published more than 500 papers. The project of "Research on Ultrasonic Motor" was awarded multiple national awards and international recognition. Based on the author and his team's research on ultrasonic motors over the last 15 years, this book has summarized their achievements as follows: Firstly, the author explains the systematic theory and design methods in using vibration and wave theory, ineluding motion mechanism, electromechanical coupling model, optimal design of structural parameters, driving, control techniques, etc. Secondly, the author creatively applies advanced analytical methods and techniques into research on the ultrasonic motors, ineluding dynamic substructure, structure dynamic modification, modal identification and separation techniques, etc. Thirdly, this book introduces many key techniques on the ultrasonic motors, ineluding an effective frequency auto-tracking technique solved by the author's team, which has been the bottleneck for application of the ultrasonic motor. The new concept of "anti-resonance/ constant current" put forward by the author is applied to the traveling wave ultrasonic motor, which can promote comprehensive performances of ultrasonic motors. Fourthly, the book shows a series of testing devices developed by the author independently or cooperatively and a series of testing methods provided by the author, which are used for various tests of parts and completed motors. Finally, this book integrates theory with application. It not only ineludes sys-
V111
Ultrasonic Motors Technologies and Ap plicalions
tematic theories and methods, but also introduces many engineering and industrial applications, such as in robots, the active flutter suppression of a two-dimensional wing, an injector for nuclear magnetic resonance, a target recognition/ tracking system, etc. In addition, the author is meticulous and precise in writing. His formula deducing, experimental data and figures are also very convincing. In summary, this book comprehensivcly and systematically describes the technologies of ultrasonic motors and their applications. It will certainly make contributions to this area. I believe the publication of this book will promote the worldwide development and practical applications of ultrasonic motors. I greatly appreciate the effort of my close friend, Prof. Zhao, in writing this wonderful book. Here I cite a Chinese poem as our mutual encouragement: "Although the stabled steed is old, he dreams to run a thousand miles". Mar 1, 2010 Bangchun Wen
Academician of Chinese Academy of Sciences Professor of :'\Iortheastern University
Preface As a new type of micro-motor, the Ultrasonic Motor CUSM) has gained rapid dcvelopmcnt and wide applications sincc thc 1980's. Unlike traditional motors with electromagnetic effect, USM is driven by ultrasonic vibration and piezoelectric effcct. This new typc of motor covcrs a wide rangc of subjects, ineluding mechanical vibration, tribology, matcrials scicncc, mechanical design, elcctronics, automatic control, super-prccision process, etc. Ultrasonic motors havc many cxccllcnt pcrformances and fcaturcs, such as simplc construction, high torquc dcnsity at low specd, dircct drive without spccd reduction gears, quick rcsponsc, better elcctromagnetic compatibility, high holding torque while power off, quict running, efficiency insensitivc to thc sizc, ctc. They have been applicd to robots, precise facilities, medical instruments, etc. With the dcvelopmcnt of new matcrials, advanced technologies, and ncw structural types, the construction and performance of ultrasonic motors will be improvcd, and their applications will bc broadencd to encompass a wider arca ineluding space vehieles, MEMS, semiconductor manufacturing, life sciences, etc. During my visit at MIT from 1992 to 1991, I started research on ultrasonic motors. I came back to China in 1991 and continued my research at Nanjing University of Aeronautics and Astronautics CNUAA). I built a research group in 1995. My group designed and manufactured a traveling wave rotary ultrasonic motor with integrated construction that operated properly by the end of that year. In 1997, I founded the Ultrasonic Motors Rescarch Center CUMRC) in NUAA. In 1999 I organised the First Chinese Workshop on Ultrasonic Motor TechnologiesCCWUMT) with the support of National Natural Sciences Foundation of China C:'\JSFC). The research and development in this area has rapidly advanced since then, and our Research Center was further promoted to be the Ultrasonic Motors Enginecring Rcsearch Center of Jiangsu Provincc in 2001. Fivc years later, the Research Center was renamed the Precision Driving Laboratory CPDLab). The 4th International Workshop on Piczoelectric Materials and Applications in Actuators(IWPMA1) was held at :'\JUAA on September 2007. For the past 15 ycars, our rcscarch team has systcmatically studied ultrasonic motors in depth and obtained considerable achievements, ineluding motion mechanism, electromcchanical coupling modcl, optimal design of structural parametcrs, driving/control techniqucs, ctc. We havc developed more than 30 typcs of ncw ultrasonic motors with indcpendent intellcctual property rights and corresponding drivcrs. Wc have 71 invention patcnts cither awarded or pending in China and more than 500 papers published in journals and conferences. Our projcct of "Rescarch on Ultrasonic Motors" was awarded multiple national awards.
x
Ultrasonic Motors Technologies and Ap plicalions
The achievements of our team can be coneluded as follows:
1. In Theory On the basis of dynamic substructure theory, a comparatively well designed electromechanical coupling model of the traveling wave type rotary ultrasonic motor is built. A new friction interface model which takes the stator teeth and the radial sliding between the stator and rotor into consideration is proposed, and this model can precisely predict the output performances of the type of ultrasonic motors. Instead of the traditional concept of "resonance point/ constant voltage", a new concept of "anti-resonance point/constant current", which is more effective for improving the efficiency and stability of the traveling wave ultrasonic motor, is put forward. An effective frequency automatic tracking method which can lower the instability of ultrasonic motor's speed Cwi thin 5 %) is found, and this method succeeds in solving the bottleneck of the ultrasonic motor Cthe speed is down while the temperature is up). A method on solving the mode mixture in the ncar frequency of the ring stator or circular plate stator is obtained, and this method can improve the stability of the ultrasonic motor. The elliptical motion equation of a bar-type traveling wave ultrasonic motor is derived, and the concept of the effective ellipse orbit which provides a theoretical basis to the optimization of the bar-type ultrasonic motors is proposed. 2. Design Methods An optimal design of structure parameters for the ultrasonic motor put forward and the corresponding software is developed. By applying the sensitivity analysis of structure parameters and structural dynamic modification technique to the design of the ultrasonic motors, an effective method which can adjust the stator's two-phase or multi-phase modal frequencies to be the same. To propose that the design of piezoelectric ceramic components used for the ultrasonic motors should be in accordance with the strain mode of stator instead of its displacement mode; To put forward a method which simultaneously utilizes different types Cextension-contraction, bending and torsion) of vibration modes in-/out-of-planes for designing all types of ultrasonic motor; To point out that the design of the flexible rotor is very important, and to present some design methods for it; To provide the concept and design principles of the step ultrasonic motors. 3. Testing Techniques A series of test devices is developed independently or cooperatively. Some effective test methods have been proposed, ineluding modal tests with nm amplitude in ultrasonic frequency area, load characteristics tests in low speed and ultrasonic frequency area, response time tests at power on/off of the ultrasonic motors, measurement devices and methods of the dynamic friction between the stator and rotor, life test equipment and methods for the ultrasonic motor, test methods of the ultrasonic motor under an extreme environmentC vacuum, high/low temperatures), and performance measurement methods and preparation devices of new friction materials.
Preface
Xl
4. Applications Two series of the ultrasonie motors (TRUM and BTRUM) have been independently developed, and some of them are applied to industry, medical and precision instruments. Moreover, we have also provided prototypes of the ultrasonic motors to some companies. This creates favorable conditions for realizing the ultrasonic motor industrialization in China. At the same time, we have investigated some precision position and constant speed control systems with multi-variable (speed, frequency and phase) using the ultrasonic motors as actuators, ineluding a position control system used for suppressing a two-dimensional wing's flutter, a constant speed control system used for injector of nuelear magnetic resonance, a composite control system based on FN:'\J and Fuzzy control strategies which is used for drivel control a robot, a control system for automatically tracking targets based on vision, a fuzzy control system applied to portable gasoline generators, etc. In addition to the achievements and innovations mentioned above, this book also fully absorbs the most advanced and important results in this area over all world in order to enrich the content. There are 15 chapters in the book. Chapter 1 is an introduction, which describes the history, elassification, characteristics and applications of ultrasonic motors. Chapter 2 describes the fundamentals of piezoelectricity and piezoelectric materials used for ultrasonic motors, and emphasizes the influence of piezoelectric materials on the performance of ultrasonic motors. The knowledge on how to select the piezoelectric materials used for USM is also introduced. Chapter 3 introduces the fundamentals of tribology and tribomaterials used for ultrasonic motors. Some tribomaterials for ultrasonic motors are proposed. In addition, the components and produce process of two new kinds of friction materials are provided. Chapter 4 introduces the fundamentals of vibration and wave applied to ultrasonic motors. It expounds the displacement and strain modes of elastic bodies such as a common rectangular, circular, ring plates, and a cylindrical shell which are used for the stator of ultrasonic motors. The strain mode is a basis of the piezoelectric component polarization division for effectively exciting the stator. Moreover, some important concepts are analyzed, such as the relation between standing wave and traveling wave, mode superposition, mode separation, and wave propagation in elastic bodies. Chapters 5-11 describe the motion mechanism, electromechanical coupling model, optimal design of structure parameters, and testing for different types' ultrasonic motors, ineluding the disk- and bar-type traveling wave ultrasonic motors, the longitudinal-torsion hybrid type ultrasonic motor, the linear ultrasonic motor, the step ultrasonic motor, the non-contact ultrasonic motor, the surface wave ultrasonic motor, etc. These chapters are the most important as they represent our academic achievements and innovations. Chapters 12-13 describe the driving and control techniques of the ultrasonic
Xli
Ultrasonic Motors Technologies and Ap plicalions
motors. Chapter 13 introduces the drive principles and design methods of the drivers in detail, and provides an actual driver circuit which is in use at PDLab. Chapter 14 introduces various tests of the ultrasonic motors, ineluding testing principles, methods, equipment, and the analysis of testing results. Chapter 15 summarizes the practical applications of ultrasonic motors and looks to the future of this area. This book is a comprehensive tutorial for practicing engineers and researchers developing the ultrasonic motor technologies and applications. It is also an up-todate reference for graduates taking a course on ultrasonic motor technologies. Finally, I tell my readers that I will greatly appreciate your comments and s ugges tions. June 5, 2010 Chunsheng Zhao
)JUAA, Nanjing, China
Acknowledgements First, I would like to thank the ='Jational ='Jatural Seiences Foundation of China, 863 High-Tech Projects, and provincial and ministerial funding projects. Many achievements described in this book are credited to these funding sources. I also express my gratitude to my colleagues who have contributed to this book. The followings people are especially thanked for writing and translating some chapters of my manuscripts: H uafeng Li, Chao Chen, Zhiyuan Yao, H ua Zhu, Ying Yang, Zhijun Sun, Weiqing Huang, Jiamei Jin, Yunlai Shi , Lin Yang, Junhui Hu, Jianhui Zhang, Qingjun Ding, Shengqiang Zhou, Yiping Wang, Yi Ding, Congyun Shi, Yubao Li, Jiantao Zhang, Wei Zheng, Hanming Peng, Xiangdong Zhao, Guiqin Wang, Wei Hu, Jian Liu, Dan Lu, Yucong Yin, Qi Chen, Ping Wang, et al. I am grateful to Prof. Bangchun Wen, Prof. Shizhu Wen, Prof. Jue Zhong, Prof. Zhiyun Shen, Prof. Liding Wang, Prof. Haiyan H u, Prof. Tieying Zhou, Prof. Chenglin Gu, Prof. Zhigang Yang, Prof. Fengquan Wang, Prof. Jinhao Qiu, Prof. Zhendong Dai, Prof. Zexiang Li, Prof. Haosu Luo, Prof. Baoku Li, Prof. Shuxiang Dong, Prof. Fei Zhou, Prof. Xiangtao Fan, Prof. Yuhong Liu, Dr. Chunning Zhang, Prof. Zhong You, et al. for their helpful comments and s ugges tions. I also want to express my heartfelt thanks to Prof. Jbrg Wallaschek, Prof. Kenji Uchino, Prof. Piotr Vasiljev, Prof. Scok-Jing Yoon, Prof. Yo shiro Tomikawa, Prof. Minoru Kuribayashi Kurosawa, Prof. Takhiro Takano, Prof. Takshi Maeno, Prof. Aydin Dogan, Prof. J ian S Dai. Dr. Toshiiku Sashida, Dr. Ichiro Ohumura, Dr. David Henderson, and Dr. Ryan Lee. They have provided the book with some papers, data, and photos via the conferences or the lectures in our PDLab. I especially thank Prof. J brg Wallaschek for writing a Foreword in the book. Last but certainly not least, I am grateful to my wife Fengying Wang, my da ugh ter Dr. Ying Zhao, my son-in-law Dr. Charles Zhou, and my grandsons Derek and J esse for their understanding and support to my work, for their care to my life, and for their encouragement to my soul.
Contents 1
Introduction··· ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 1
1. 1 History of Ultrasonic Motors ................................................ 2 1. 2 Characteristics of Ultrasonic Motors and Their Classification'" ...... 7 1. 2. 1
Characteristics of Ultrasonic Motors
1. 2. 2
Classification of Ultrasonic Motors'" ... ...... ... ...... ... ...... ... ... 8
...... ... ... ... ... ... ... ... ... ... 7
1. 3 Comparison with Electromagnetic Motors
.............................. 12
1. 3. 1
Load Characteristics ................................................... 13
1.3.2 1.3.3
Transient Response Characteristics
Energy Transform of Motors and Their Micromation
............ 13
................................. 14
1. 4 Applications and Development Trends of Ultrasonic Motors 1.4. 1
15
Applications .. · .. · .. · .. · .. · .. · .. · .. · .. · .. · .. · .. · .. · .. · .. · .. · .. · .. · .. · .. · 15
1.4.2 Development Trends ................................................... 15 References ........................................................................... 18
2
Fundamentals of Piezoelectricity and Piezoelectric Materials for Ultrasonic Motors ............................................................... 21
2. 1 Development and Classification of Piezoelectric Materials ............ 21 2.1. 1
Historical Development of Piezoelectric Materials .................. 21
2. 1. 2
Classification of Piezoelectric Materials'" ...... ... ...... ... ...... ... 22
2. 2 Electrical Properties of Piezoelectric Materials
... ... ...... ... ...... ... 23
2. 2. 1
Dielectric Properties and Dielectric Loss
...... ... ...... ... ...... ... 23
2. 2. 2
Ferroelectric Properties and Polarization
...... ... ...... ... ...... ... 25
2. 3 Properties and Constitutive Equations of Piezoelectric Materials
27 ........................... 27
2. 3. 1
Elastic Properties and Their Codficients
2. 3. 2
Piezoelectric Properties and Piezoelectric Equations
............... 28
2. 4 Vibration Types of Piezoelectric Vibrators .............................. 31 ............ 31
2.1. 1
Piezoelectric Vibrators and Their Equivalent Circuits
2. 4. 2
Characteristic Frequencies of Piezoelectric Vibrators ............... 33
2. 4. 3
Coupling Coefficient and Quality Factor
...... ... ...... ... ...... ... 35
2. 5 Applications of Piezoelectric Materials to Ultrasonic Motors 2.5. 1
Piezoelectric Ceramics Used for Ultrasonic Motors
2.5.2
Applications of Piezoelectric Materials to Other Actuators
38
............... 39 41
2. 6 Advances in Novel Piezoelectric Materials .............................. 45
Ultrasonic Motors Technologies and Ap plicalions
XVI
References
3
........................................................................... 17
Fundamentals of Tribology and Tribomaterials for Ultrasonic Motors ........................................................................... 50
3. 1 Basic Tribology
............................................................... 51
3. 1. 1
Surface of Tribomatcrials ............................................. 51
3.1.2
Friction and Its Classification
3. 1. 3
Friction Mechanism ................................................... 53
3.1.4
Wear Mechanism ...................................................... 56
3. 1. 5
Wear Surface for Stator and Rotor of Ultrasonic Motors
....................................... 52
3.2 Tribomaterials Used for Ultrasonic Motors
58
........................... 58
3.2. 1
Basic Requirement, Classification and Selection Principle ......... 58
3. 2. 2
Influence of Composition on Tribological Properties ............... 60
3. 2. 3
Preparation of Tribomaterial .......................................... 62
3. 3 Influence of Tribomaterials on Performance of USM .................. 67 3.3. 1
Influence of Elastic Modulus and Hardness
3. 3. 2
Influence of Friction Coefficient
3. 3. 3
Influence of Anisotropy
........................ 67
...... ... ...... ... ...... ... ...... ... 69
...... ... ...... ... ...... ... ...... ... ...... ... 70
3. 4 Friction Testing for Tribomaterials ....................................... 72 3.1. 1
Quasi-static Friction Testing .......................................... 72
3.4.2
Dynamic Friction Testing ............................................. 73
References
4
........................................................................... 71
Fundamentals of Vibration for Ultrasonic Motors ........................ 76
1. 1 Natural Vibration of Elastic Body··· .................................... 76 4. 1. 1
Longitudinal Vibration of Bars ....................................... 77
1. 1. 2
Characteristics of Natural Modes .................................... 78
4. 1. 3
Torsional Vibration of Shafts
4.1.4
Bending Vibration of Beam
.......................................... 81
... ...... ... ...... ... ...... ... ...... ... 80
1. 1. 5
:"Iatural Vibration of Plates
.......................................... 82
4.1. 6
:"Iatural Vibration of Cylindrical Shells .............................. 92
4. 2 Forced Vibration of Elastic Body··· ....................................... 95 1. 2. 1
Response of PZT Bar to Distributed Electric Field
... ... ...... ... 96
4.2.2
Metallic Bar Excited by Single or Multiple PZT Pieces ............ 99
1.2.3
Response of Beam to Constant Electric Field Intensity
4.2.4
Excitation of Simply Supported Beam by PZT Pieces ............ 101
1.2. 5
Response of Thin Plate to PZT Piece Excitation .................. 106
101
1. 3 Wave Propagation in Elastic Body··· .................................... 107 ............................................. 107
4.3.1
Basic Concept of Wave
1.3.2
Waves in Elastic Body··· .......................................... 108
4.3.3
Superposition of Waves
............................................. 110
Contents
XVll
1. 3. 1
Formation of Traveling Waves
4.3. 5
Formation of Elliptical Trajectory··· .............................. ll4
... ...... ... ...... ... ...... ... ...... III
References ........................................................................... 115
5
Operating Mechanism and Modeling of Traveling Wave Rotary Ultrasonic Motor ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 118 ....................................... ll9
5. 1 Operating Mechanism of TRUM
.................. ll9
5.1.1
Structure and Operation Mechanism of TRUM
5.1.2
Formation of Traveling Wave in Stator
5. 1. 3
Elliptical Motion Trajectory of Surface Points on Stator ......... 123
5. 1. 1
EHect of Amplitude and Phase on Elliptical Motion··· ...... ... ...... 121
........................... 121
5. 1. 5
Polarization Pattern of Piezoelectric Ceramic Components
5. 1. 6
Three-dimensional Motion Analysis of Points on Stator ......... 129
128
5. 2 Semi-Analytical Electromechanical Coupling Model of Stator ...... 132 5. 2. 1
Substructure Division of Stator
.................................... 133
5. 2. 2
Characteristic Matrix of Substructures a and b
5.2.3
Characteristic Matrix of Substructure c··· ........................ 139
... ...... ... ...... 133
5.2.4
Electromechanical Coupling Model of Stator
5.2. 5
Computation Example of Dynamic Characteristics of Stator ...... 141
..................... 110
5. 3 Contact Model Between Stator and Rotor .............................. 111 5.3. 1
Interface Assumptions
............................................. 145
5.3.2
Interface Force and Power Transmission ........................... 146
5.3.3
Interface Energy Loss and Power Transmission Efficiency··· ... 118
5.3.4
Contact Model Between Stator and Rotor
5.3. 5
Contact InterLace Simulation
........................ 149
....................................... 150
5.4 Electromechanical Coupling Model of TRUM and Its Simulation ...... 152 5.1. 1
Electromechanical Coupling Model of TRUM ........................ 152
5.4.2 Performance Simulation of Ultrasonic Motor ..................... 151 References ........................................................................... 158
6
Design and Manufacture of Traveling Wave Rotary Ultrasonic Motors ........................................................................... 161
6. 1 General Design Process of TRUMs
.................................... 161
6.1. 1
Structure Sizes of the Stator
6. 1. 2
Design of Rotor Size
....................................... 162
6.1.3
Choice of Materials ................................................... 166
................................................ 165
6. 2 Operating Modes of Stator and Polarization of PZT Ring 6.2. 1 Design of Modal Frequency .0
6.2.2
••
0
••
0
••
0
••
0
••
0
••
••
0
••
0
••
0
••
0
••
0
••
0.
167 168
Polarization of Piezoelectric Ceramics .............................. 168
6. 3 Structure Form of Stator and Its Modal Analysis 6. 3. 1
0
Structure Form of Stator
... ...... ... ...... 170
...... ... ...... ... ...... ... ...... ... ...... 170
Ultrasonic Motors Technologies and Ap plicalions
XV111
6. 3. 2
Modal Analysis of Stator
...... ... ...... ... ...... ... ...... ... ...... 170
6.4 Sensitivity Analysis and Avoiding of Mode Mixture of Stator ...... 172 6.1. 1
Principle of Sensitivity Analysis .................................... 172
6.4.2
Sensitivity Analysis of Stator for TRUM-60
6. 1. 3
Mode Separation of Stator for TRUM-60
..................... 173 ...... ... ...... ... ...... 174
6. 5 Optimal Design of Stator ................................................... 176 6.5. 1
Optimal Model of Stator ............................................. 176
6. 5. 2
Example of Optimal Design of Stator .............................. 177
6. 6 Adjustment of Two Phase Modal Frequencies of Stator ............ 179 6. 6. 1
Method of Adjusting of Two Phase Modal Frequencies
180
6. 6. 2 Example of Adjusting of Two Phase Modal Frequencies 182 6. 7 Analysis of Flexible Rotor ................................................ 183 6.7. 1
Importance of Rotor's Flexibility for Performance of Motor
185
6.7.2
Comparison of Contact Area of Rigid and Flexible Rotor
188
6.7.3
EHect of Rigid and Flexible Rotors on Mechanical Characteristics
189
6.7.4
Design and Manufacture of Flexible Stator ........................ 189
6.8 Manufacturing Techniques of TRUM ................................. 191 References ........................................................................... 193
7
Bar-type Traveling Wave Rotary Ultrasonic Motors ..................... 195
7. 1 Review of Bar-type Ultrasonic Motor
................................. 195
7. 2 Construction and Motion Mechanism of SDOF Motor ............... 196 7.2. 1 Construction ......................................................... 196 7.2.2
Motion Mechanism ................................................... 197
7. 3 Optimal Design for SDOF Motor
....................................... 201
7.3. 1
Design Principle ...................................................... 201
7.3.2
Dynamic Model
7.3.3
Sensitivity Analysis
7.3.4
Objective Function ................................................... 208
7.3. 5
Optimal Algorithm and Results
...................................................... 203 ................................................ 207 .................................... 208
7.3.6
Modal Frequency Modification of Stator ........................... 210
7. 3. 7
Design of Flexible Rotor ............................................. 213
7. 1 Performance Simulation for SDOF Motor .............................. 211 7. 4. 1
Dynamic Model
...................................................... 214
7.1.2
Contact Analysis
7.4.3
Performance Simulation ............................................. 216
................................................... 215
7.5 Motion Mechanism of 3-DOF Motor .................................... 219 7.5.1
Construction and Operating Modes ................................. 219
7.5.2
Motion Mechanism ................................................... 219
7. 6 Optimal Design of Stator of 3-DOF Motor 7.6. 1
Construction and Objective Function
........................... 221 .............................. 221
Contents 7.6.2
XIX
Optimal Algorithm and Results
.................................... 223
7. 7 Performance Measurement of 3-DOF Motor ........................... 225 7.7. 1
Testing Equipment and Results .................................... 225
7.7.2
Effect of Pre-pressure on Mechanical Performance ............... 226
7. 8 Driving and Control Techniques of 3-DOF Motor
.................. 227
.............................. 227
7.8. 1
Configuration of the Control System
7.8.2
Control for Trajectory Tracking .................................... 227
References······ ... ...... ... ...... ... ...... ...... ... ...... ... ...... ... ...... ... ...... 229
8
Ultrasonic Motor Using Longitudinal-Torsional Hybrid Vibration
8. 1 Current Research of LTUM 8. 2 Multi-mode Type LTUM
... 232 ... ...... ... ...... ... ...... ... ...... ... ...... 232
...... ...... ... ...... ... ...... ... ...... ... ...... 235
8. 2. 1
Motion Mechanism ................................................... 235
8.2.2
Structure Design of Multi-mode Type LTUM ..................... 239
8. 3 Contact Model between Stator and Rotor .............................. 213 8. 3. 1
Modeling of Contact Interface··· ... ...... ... ...... ... ...... ... ...... 243
8. 3. 2
Friction Loss on Interface and Efficiency of LTUM
8.3.3
Simulation of Performance of LTUM .............................. 219
... ... ...... 218
8. 4 Mode Conversion Type Ultrasonic Motor .............................. 255 .................................... 255
8.1. 1
Structure and Operation Modes
8.4.2
Principle of Mode Conversion ....................................... 256
8.1.3 8.4.4
Design of Mode Conversion Type LTUM with Holes ............ 258 Testing ............................................................... 259
References ........................................................................... 262
9
Linear Ultrasonic Motors
.................................................. . 265
9. 1 State of the Art of Linear Ultrasonic Motors
266 9. 2 Linear Ultrasonic Motors Based on d'l Effect ........................ 270 9.2. 1
Linear Ultrasonic Motor with Double Driving Feet ............... 270
9.2.2
Linear Ultrasonic Motor with Single Driving Foot ............... 278
9. 3 Linear Ultrasonic Motors Based on d" Effect ........................ 281 9. 3. 1
Linear Ultrasonic Motor with Butterfly Shaped Stator
9. 3. 2
Linear Ultrasonic Motor with Wheel Shaped Stator
9.4 Contact Model of Standing Wave Type LUSMs 9.1. 1
Steady State Characteristics
282 ... ... ...... 288
..................... 292
....................................... 292
9.4.2
Transient Responses
................................................ 293
9.1.3
Simulation Examples
................................................ 294
9. 5 Synergetic Operation Technique of LUSMs ........................... 296 References ........................................................................... 298
Ultrasonic Motors Technologies and Ap plicalions
xx
10
Step Ultrasonic Motors ...................................................... 300
10. 1 Step Control of USM ...................................................... 301 10. 1. 1 10.1.2
Startup and Shutdown Characteristics of USM ............... 301 Step Control for USM ............................................. 303
10. 1. 3 Factors Impacting on Single-step Positioning Accuracy 308 10. 2 Step USM with Fixed Step length ....................................... 309 10.2. 1 10.2.2
Standing Wave USM Used for Constructing Step USM 309 Modal Rotary Type Step USM ................................. 312
10.2.3
Self-correction Peak Type Step USM
........................... 322
References······ ... ...... ... ...... ... ...... ...... ... ...... ... ...... ... ...... ... ...... 325
11
Other Ultrasonic Motors
................................................... 327
11. 1 )Jon-Contact Type Ultrasonic Motors ................................. 327 11. 1. 1
Classification and Development
11. 1. 2
Operating Principle ................................................ 331
...... ... ...... ... ...... ... ...... 328
11. 1. 3
Design of Non-contact USM
11. 1. 1
Performance Measurement of Non-contact USM ............... 336
... ...... ... ...... ... ...... ... ...... 335
11.1. 5
Design of Non-contact Type USM with Disk Stator
11. 1. 6
Testing of :'\fon-contact USM with Disk Stator
11. 2 Linear Surface Acoustic Wave Motor
337
...... ... ...... 338
................................. 310
11. 2. 1
State of the Art ................................................... 340
11. 2. 2
Surface Acoustic Wave and Its Generation
11.2.3
Operation Mechanism ............................................. 316
..................... 343
References······ ... ...... ... ...... ... ...... ...... ... ...... ... ...... ... ...... ... ...... 349
12
Driving Techniques for Ultrasonic Motors .............................. 351
12.1 Design Requirements for Drivers ....................................... 351 12. 2 Signal Generator ............................................................ 353 12.2. 1
RC Multivibrator
................................................ 353
12.2.2
555 Multivibrator
................................................ 351
12. 2. 3
Voltage Controlled Oscillator .................................... 355
12. 3 Frequency Divider and Phase Splitter ................................. 356 12.3. 1
FDPS Composed by Shift Register .............................. 356
12.3.2
FDPS Composed by CPLD ....................................... 358
12.1 Power Amplifier Techniques ............................................. 359 12. 5 Electrical Characteristics of Ultrasonic Motors
..................... 362
12.5. 1
Experimental Results and System Description .................. 362
12.5.2 12.5.3
Analysis of Vibration States and Driving Method ............ 361 Experimental Results ............................................. 364
12. 6 Influence of Matching Circuit on Performance of Driver 367 12.6. 1 Influence of Matching Capacitor ................................. 368
Contents
XX!
12.6.2
Influence of Matching Inductors ................................. 371
12.6.3
Influence of USM on Driver
.................................... 373
12.7 :'\Jon-transformer Driver with Resonance Voltage Step-up ......... 376 References······ ... ...... ... ...... ... ...... ... ...... ... ...... ...... ... ...... ... ...... 383
13
Control Techniques for Ultrasonic Motors
.............................. 385
13. 1 Classification of Control for Ultrasonic Motors ..................... 385 13. 2 Speed Adjusting Mechanism and Control Methods of USM ...... 387 13.2. 1 Voltage Amplitude Adjusting .................................... 387 13.2.2
Frequency Adjusting
13.2.3
Phase Difference Adjusting ....................................... 389
............................................. 388
13. 3 Stability Control Techniques for Ultrasonic Motor 13.3. 1
............... 390
Principle of Frequency Automatic Tracking ..................... 390 .......................................... 392
13.3.2
Detection of Amplitude
13.3.3
Implementation of FAT System ................................. 392
13.4 Ultrasonic Motors Used as Servo Motors 13.1. 1
Ideal Servo Actuator- USM
........................... 393
.................................... 393 .................. 391
13.4.2
Requirements of Servo Control Using USM
13.4.3
Servo Control System Using USM .............................. 395
13.1.1
PID Controller Using USM ....................................... 397
13.4. 5
Adaptive Controller Using USM ................................. 404
13.1. 6 Fuzzy Controller Using USM .................................... 411 References ........................................................................... 117
14
Testing Techniques for Ultrasonic Motors
... ... ...... ... ...... ... ...... 119
14. 1 Modal Testing for Parts and Assemblies .............................. 419 11. 2 Measurements of Pre-pressure .......................................... 123 11. 3 Measurement of Transient Characteristics ........................... 121 14. 3. 1
Testing Principle··· ...... ...... ... ...... ... ...... ... ...... ... ...... 424
11. 3. 2
Transient Characteristics of USMs .............................. 126
14.4 Measurement of Load Characteristics ................................. 427 11. 1. 1
Measurement System for Load Characteristics··· ...... ... ...... 427
11. 1. 2
Measured Results for TRUM-60 Load Characteristics ......... 129
14. 5 Environmental Testing for Ultrasonic Motors
..................... 430
14.5.2
High/Low Temperature Environmental Testing ............... 130 Vacuum Environment Testing .................................... 132
11.5.3
Load Characteristics of USMs in Vibration Environment······ 434
11.5. 1
11.5.1
Load Characteristics of USMs under Strong Shock ............ 137
14. 5. 5
Test and Analysis of :'\roise from Ultrasonic Motors
438
11.5. 6 Testing of USMs in Hygrothermal Environment ............... 110 11. 6 Life Testing for USMs ................................................... 112
Ultrasonic Motors Technologies and Ap plicalions
XXll
11.6. 1
Design 01 Life Testing System
................................. 112
Life Testing Results and Analysis for TRUM .................. 444 References ........................................................................... 115
14. 6. 2
15
Applications of Ultrasonic Motors in Engineering ..................... 448
15. 1 Applications in Domestic Engineering ................................. 119 15.1. 1
Application in Camera ............................................. 449
15.1.2
Application in Cell Phone
15.1.3
Application in Watch
....................................... 151
............................................. 151
15. 2 Applications in Industrial Engineering ................................. 452 .............................. 152
15.2. 1
Application in Gasoline Generator
15.2.2
Applications in Automobile ....................................... 153
15.2.3
Applications in Robot ............................................. 156
15.2.1
Application in Surveillance Camera PlatIorm
15.2. 5
Applications in Precision Positioning Stage ..................... 159
15. 3 Applications in Biological and Medical Engineering
.................. 458 ............... 461
15.3. 1
Applications in Medical Facility··· .............................. 161
15.3.2
Applications in Biomedical Engineering
15.4 Applications in Aerospace Engineering 15.1. 1
Applications in Aircraft
15.4.2
Applications in Aerospace
........................ 162
.............................. 464
.......................................... 161 ....................................... 466
References ........................................................................... 168
Index ....................................................................................... 469 Appendix A
Natural Vibration Frequencies and Mode Shape Functions of Bars Shafts, Beams, and Plates ... ... ... ... ... ... ... ... ... ... ... 477
Appendix B
Natural Vibration Mode Shapes of Bars, Shafts, and Beams .............................................................................. 179
Appendix C
Natural Vibration Displacement and Strain Mode Shapes of Plates, and Their Nephogram .................................... 187
Symbols The following symbols arc commonly used with the attached definitions. unless otherwise specified in the text. .1':.y.z
Spatial coordinates in a global system
u,v,w
Displacements in
.1':.
U o .Vo .Wo
Ampli tudes in
y. z directions
U o ,Va ,Wo
Displacements of neutral layer
u,v,w
V eloci ties in
u,v,w
Accelerations in
.1':.
.1':.
y. z directions
y. z directions .1':.
y. z directions
Tangential velocity Rotating speed (Speed) Length Width
b
h
Height. thickness
r(r, )
Radius
D(d)
Diameter
m
Mass
Mi
i th modal mass
Ki
i th modal stiffness
Fi
i th modal force
Ci
i th modal darning
S
Area
P E
Density of material
/1
Poisson ratio
G
Shear modulus of elasticity
lUx .Iy .Ie .Ip)
Inertia moment
A k
Wave length
Young's modulus
V clocity of wave propagation Wave number
/1 (/1d •/1, )
Friction coefficient
u(ai • Til )
Stress matrix
(Ci )
F( Fn.F,)
Strain matrix External force
Ultrasonic Motors Technolof{ies and Ap plicalions
XXIV
Friction force
Fr f(x) ,fey) ,fez)
Distributed forces in x, y, z directions
P(P o )
Pre-pressure (Preload)
M(M" Mx,
MT
M~)
Bending moment Torque Temporal variable Voltage sign function of the polarization of phase A Voltage sign function of the polarization of phase B Code of the tooth cell
e
Shape function of annular cell Mass matrix of annular cell /j"
Displacement column matrix of substructure a
M"
Mass matrix of substructure a
K"
Stiffness matrix of substructure a
M'
Mass matrix of tooth e
a'
Displacement column matrix of nodes of tooth e Displacement column matrix of inner nodes of tooth e
r
Condensed matrix of tooth e
K'
Condensed stiffness matrix of tooth e
T
Kinetic energy
V
Potential energy
f
Frequency (Friction force) Angular frequency
W
¢n (x)
,
¢n
nth mode shape
fn (W n ) c;p
nth mode frequency
cp( cpn)
Phase angle
q(t)
Mode shape matrix Modal coordinate Bending mode Mechanical quality factor Force coefficient matrix Shape function matrix Variable for structure design Sensitivity with respect to p; Relative sensitivity with respect to Pi Radial shape function matrix Node column matrix of annular cell Stiffness matrix of annular cell Displacement column of substructure b Mass matrix of substructure b Stiffness matrix of substructure b
Symbols
xxv
K'
Stiffness matrix of tooth e
aj
Displacement column matrix of boundary nodes of tooth e Static condensed matrix of tooth e Condensed mass matrix of tooth e
M'
Dielectric constant matrix under
E=
constant
Lagrange function Variational Work Charge on electrode Generalized coordinate column matrix K,
Generalized stiffness matrix
e,
Radial unit vector Axial unit vector Circumferential unit vector
PCP)
Polariza tion in tensi ty vector
E( Ei )
Electric field intensity vector
D( D i
Electric displacement vector
)
S(5,)
Strain tensor
TCTi)
Stress tensor
(Sij)
S
Flexibility coefficient matrix
c (e i})
Stiffness coefficient matrix
k
Electromechanical coupling coefficient
dedi} )
Piezoelectric constant matrix
im in i, i, i,
Minimum impedance frequency Series resonance frequency
i
p
Parallel resonance frequency
(Ei)
Dielectric constant matrix
Maximum impedance frequency Resonance frequency Anti-resonance frequency
v
Voltage
V pp
Peak-peak value of voltage
Vo VA,VB
Voltage amplitude
I (i. io )
Current matrix
Vol tage of phase A or B
I
Unit matrix
R(R l .Rd .Rm)
Resistance
C(Ca ,Cl .Cm ) L(L .Lm .Lp .L,)
Capacitance
Y
Admittance
j
Inductance
XXVI
Ultrasonic Motors Technolof{ies and Ap plicalions
Z(Zn,)
Impedance (Mechanical impendance)
W(W k )
Work (electric potential energy)
D
Duty cycle
1']
Integral time constant
Til
Differential time constant
K]
Integral coefficient
Kn Kp
Differential coefficient
VI
Isolated electrode voltage
Scale factor
Chapter 1
Introduction Traditional motors based on the electromagnetic principle have been invented and developed for more than one hundred years. As actuators and power sources, the motors have been widely used in many fields all over the world and have made a great contribution to our society. Over the years, the theories, design methods and manufacturing technologies of traditional motors have been developed so successfully that little improvement can be made to them.
However, due to ad-
vanced science and technology, especially in hi-tech products such as spaceships, satellites, launch vehicles, various electronic equipment, and precision instruments, many new requirements for motors have been raised, including a small size, light weight, low noise, no electromagnetic interference, etc. Due to limitations on the principle and structure, traditional motors are difficult to meet these requirements. Many countries in the world strive to explore various new, small, and special motors such as electrostatic motors, ultrasonic motors (USMs) , bionic motors, photo-thermal motors, shape memory alloy motors, microwave motors, etc. As integrated hi-tech products, the new, small, and special motors apply a variety of new technologies, including computers, automatic control, precision machinery, new material and modern manufacturing. They are increasingly becoming indispensable devices not only in the development of aerospace equipment, but also in the achievement of industrial automation, office automation and home automation. The ultrasonic motor is relatively mature among these new, small, and special motors. USMs have been developed as a new concept of motors since the 1980's. It utilizes the vibration of the elastic body (stator) in the ultrasonic frequency band and the reverse piezoelectric effect of piezoelectric materials. The mechanical movement and torque are obtained by means of the frictional contact force between the stator and rotor or slider. USMs can meet many new requirements for small and special motors because of advantages such as small size, light weight, compact structure, fast response, low noise, and no electromagnetic interference. )Jowadays, this kind of motors is being developed very quickly and applied to more and more fields. As an introduction to the book, this chapter summarizes the history, features, classification and applications of ultrasonic motors.
C. Zhao, Ultrasonic Motors © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011
2
1.1
Ultrasonic Motors Technologies and Ap plicalions
History of Ultrasonic Motors L1J
In 1940s, scientists discovered the ceramic material of BaTi0 3
•
which is easy to
be processed and can be made into piezoelectric elements of special shape and can be poled in arbitrary direction. This discovery greatly promoted the development of piezoelectric actuator technology. As early as in 1948, Williams and Brown applied for the first patent of "piezomotor" in history-2-, whose structure is shown in Fig. 1. 1. Their invention reveals the basic tenets and creates a new period of ultrasonic motor. However, ultrasonic motors were not rapidly developed because of the limitations of materials and processing technology at that time. Man's attempt to use vibration of elastic bodies to obtain the power began with the clock and watch industry. In 1960, Swiss Bulova watch Co., Ltd. used the reciprocating displacement of a metal tuning fork to drive watch gears: 3 : , as shown in Fig. 1. 2. This clock's operating frequency is 360Hz. This watch has an error of one minute per month. which created a record at that time. About ten years later, roughly in 1970-1972, Siemens and Matsushita Electrical Industries developed a kind of linear actuator and step motor, which used the key component such as piezoelectric vibrator. Because the piezoelectric vibrator's resonant frequency is as high as tens of thousands hertz and vibration amplitude is very small. the motor can not obtain a large torque or power. Drive coil
Drive coil Vibration direction
Fig. 1. 1
Sta tor of the first ultrasonic motor
Fig. 1. 2 Driving mechanism of tuning fork watch
In 1965. Lavrinenko from the Soviet Union designed a kind of ultrasonic motor, as shown in Fig. 1. 3, which used the vibration of piezoelectric plate to drive the rotor. He was granted an invention patent: 4J and summed up the characteristics of the ultrasonic motor: simple structure, low cost, low speed, high torque, large energy density, high precision, and high energy conversion efficiency. In 1973, Barth in IBM proposed a structural design scheme with the principle of modern ultrasonic motor[3: , as shown in Fig. 1. 4. He used two piezoelectric actuators to produce longitudinal vibration of horns. The contact friction between the rotor surface and horns end drives the rotor. In 1975, Vishnevsky also proposed a design scheme similar to Barth' S[6 J • They used a spring to press the edge
Chapter 1
Introduction
3
of a rectangular piezoelectric composite stator, which excited a longitudinal bra tion modc to drive the rotor, as shown in Fig. 1. 5.
Vl-
Rotor Output shaft
P iezoelectTi c
actuator 2
Piezoelectric
actu ator I
Fig. 1. 3
Lavrinenko's USM
Fig. 1. 4
Barth's design of USM
Excitat ion i b'l131
Fig. 1. 5
Vishnevsky's design of USM
In 1981, Lithuanian Vasiliev successfully developed an ultrasonic motor with the ability of driving larger loads L7J , as shown in Fig. 1. 6. The stator of this motor is a Langevin vibrator, which excites the longitudinal vibration mode of the metal shect contacting with thc rotor that is driven by thc friction force bctwcen the sheet and rotor. This structure of the motor can decrease the operating frequency and amplify thc vibration amplitude at the samc time. Thc usc of such motors to thc wheel of gramophonc becamc the first practical application of piczoelectric actuator in that cra. Aftcr Vasiljev's rescarch findings, Sashida in 1982 dcsigncd and made a standing wavc ultrasonic motor:"], as shown in Fig. 1. 7. This motor uscd a Langcvin vibrator. Its driving frequency was 27. 8kHz, input electric power 90W, output mechanical power SOW, output torque O. 5:'\J-m, output rotational speed 2800 r/ min, and the efficiency 60 %. It can be said that this piezoelectric ultrasonic motor met the performance requirements for actual applications at the first time. Because this motor's metal film and the rotor were fixed at the same position, serious wears existed on the contact surfaces. To solve this problem, in 1983 Sashida designed and manufactured another traveling wave ultrasonic motor, and in 1985 was granted a patent: 9: in USA, as shown in Fig. 1. 8. This motor realizcd the rotor rotation through thc traveling wavc instcad of the standing wavc. The formcr drivcs thc rotor continuously, while the lattcr drives the rotor discontinuously, so thc abrasion on the contact
Ultrasonic Motors Technologies and Ap plicalions
4
surface is decreased uSing the traveling wave. The successful development of such motors paved the way towards practical applications of ultrasonic motors. In the same year, Sashida put forward two design schemes of traveling wave linear ultrasonic motors(LUSMs) based on the same principle llo -: one is straight beam type, as shown in Fig. 1. 9, and the other is ring beam type, as shown in Fig. 1. 10. Piezoccralllic Metallic
Rotor Beari ng
Flexible sheet
Fig. 1. 6
Vasiliev's structure of USM
Fig. 1. 7
Sashida's standing wave USM
Rotor
,
: ~ Moving direction of rotor
)- - - - ~ ~...-,.-;..- - - - - - - - - - - - -: ;;;>.,.,;-<"
.........
(a) Principle
(b) Construction
Fig. 1. 8
Sashida's traveling wave USM
Ring beam type vibrator Pre-pressure
lJl:ii=l1I:::,..- Piezoelectric ceramic
---------------
II
Piezoelectric ceramics
Fig. 1. 9 Sashida's design of ring beam type linear motor
Fig. 1. 10
Sashida's straight beam type
linear motor
In 1985, Kumada developed a longitudinal-torsional hybrid ultrasonic motor that can be driven by single phase signaJ=ll]: maximum output torque is 1. 3N-m, rotatory speed is 120r/min. In 1987, Ishc in Panasonic, Inc. designed a ringtype traveling wave ultrasonic motor based on Sashida's traveling wave motor]2J. The tooth structure of this motor's stator magnified the amplitude of the
Chapter 1
Introduction
5
stator, which greatly improved operational efficiency. In the same year, Sashida founded a company called Shinsei corporation and began to commercialize ultrasonic motors. Canon Co., Ltd. first used the ring-type ultrasonic motor in the zoomar of EOS camera in 1987. It indicated that the development of ultrasonic motors goes into a period of engineering applications.
Since ultrasonic motors developed by Sashida were commercialized, this new concept motor aroused the interest of researchers. Many various USMs with different exciting principles, oped
structures,
and performances have been devel-
113-l1_.
In China, research on ultrasonic motor started in 1990s, and Tsinghua University, Zhejiang University, Jilin University, Nanjing University of Aeronautics and Astronauties()JUAA), and Harbin Institute of Technology, began with the research on ultrasonic motor in turn and got many achievements. In 1989, Tieying Zhou and Shuxiang Dong have been granted the first Chinese invention patent on ultrasonic motor: l5 ] and obtained a series of research results on miniature ultrasonic motors. From 1992 to 1991, the author was in MIT as a visiting professor and participated in the joint research and development of ultrasonic motors at the Departments of Aeronautics and Astronautics Engineering and Computer and Electrical Engineering. After returning to China, the author continued to engage in the research on and development of ultrasonic motors. A disk-type traveling wave ultrasonic motor, was successfully developed at the end of 1995. The author led Precision Driving Laboratory(PDLab) of )JUAA to carry out a systematic and indepth study on new ultrasonic motor operating mechanism, electromechanical coupling dynamics model, structural parameter optimization, driving and control techniques, and test techniques of new USMs. We have published more than 500 research papers, and developed more than 30 kinds of ultrasonic motors and drivers, for which we have 71 invention patents awarded and pending in China. Some of ultrasonic motors have been applied to scientific researchers and commercial products in China. Table 1. 1 lists specifications of TRUM series of traveling wa ve ultrasonic motor. Fig. 1. 11 shows a part of ultrasonic motors and drivers developed by PDLab and color Fig. 1 introduces other particular ultrasonic motors developed by PDLab and application examples. In recent years, linear ultrasonic motors and precise positioning stages driven by these linear motors arc also being developed by PDLab.
6
Ultrasonic Motors Technologies and Ap plicalions Table 1. 1
Specifications of TRUM series of traveling wave ultrasonic motors
Type
TRUM-30 TRUM-10 TRUM-60 TRUM-N ' TRUM-80 TRUM-I00
Rotor diameter/mm
30
10
60
60
80
100
Operating frequency/kHz
35
40
41
41
36
30
Drive voltage/V
100
100
100
100
150
200
Rated torque/()lom)
o.
o.
2
0.5
0.5
1.5
2. 1
150
100
100
80
50
3
5.0
5.0
12
12
Rated rotary speed/ (r/min)
10
180
Rated output powcr/W
o.
Max. rotary speed/ (r/min)
250
200
180
180
100
60
Stall torque/()lom)
0.2
o. 1
1.2
1.2
2.5
3.5
>0.2
>0.4
>1.2
>1.2
>2.5
>3.5
Self-locking torque/ ()10m)
18
Rotary direction
CW, CCW
Start response time/ ms
<1
<2
<3
<3
<6
<10
Shutdown response time/ ms
<1
<1
<1
<1
<1
<1
Input voltage/V
DC15+1
DC15+ 1
DC15+ 1
DC15+1
DC15+1
DC15+1
Motor weight/g
10
60
250
250
680
780
-x- No magnetic.
I
(a) TRUM series oflraveling wave USM
(c) Driver fo r TRUM
Figo 1. 11
(b) BTRUM series o ftl"3ve ling wave USM
(d) Driver for BTRUM
A part of ultrasonic motors and corresponding drivers developed by PDLab
Chaptcr 1
7
Introduction
Characteristics of Ultrasonic Motors and Their Classification
1.2
Ultrasonic motors based on the new principle and structure have many advantagcs comparcd with traditional motors based on elcctromagnctic effcct and it promises the prospect for broad applications since their birth.
1. 2. 1
Characteristics of Ultrasonic Motors
1. Advantages (1) Compact structure, dcsign flcxibility, large torque density (torquc/ weight
ratio): Ultrasonic motors have thc advantagcs of compact structure and flexiblc dcsign bccausc piezoelectric componcnts can cxcitc diffcrent types of vibration, ineluding longitudinal, bending, and torsional vibrations. As shown in Table 1. 2, thcir torque dcnsity can be 3-5 times of traditional motor's. (2) High torque at low speed, can directly drive loads without gear: Positioning accuracy and response speed are greatly enhanced because this advantage reduces additional volume and weight caused by the gear box, vibrations and noise, energy loss and position error caused by transmission. (3) Motor's moving parts (rotor) with the features of small inertia, fast responsc (microsccond level), self-locking, and a high holding torque: Ultrasonic motors can arrive at stable speed in several milliseconds and brake even faster becausc of friction betwecn the stator and rotor. Table 1. 2
Thc comparison bctwccn clcctromagnctic and ultrasonic motors
Motor classification
Manufac-
Stall torque
turcrs
/CN'm)
Rotary speed Maximum Weight Torque density efficiency without load /C)I'cm/g) /g /Cr/min) /%
EM, DC, Brush
Micro Mo
O. 003 32
13 500
11
O. 030 2
71
EM, DC, Brush
Maxon
0.012 7
5 200
38
O. 033 4
70
EM, DC, Brush
Mabuebi
0.015 3
14 500
36
O. 042 5
53
EM, DC, Brusb
Aeraflex
O. 009 88
4000
256
O. 003 86
20
Astra
0.075 5
11 500
310
O. 022 2
20
EM, Alternating voltage / current, Three phases
USM, Standing wave, Kumada Longitudinal-torsional type
1. 331
120
150
0.889
80
USM, Traveling wave type, 060
Shinsei
1.0
150
260
0.385
35
USM, Traveling wave type, 060
PDLab
1.2
180
250
0.522
30
8
Ultrasonic Motors Technologies and Ap plicalions
(1) Good controllability of position/ velocity and high resolution of displacement: Ultrasonic motors can achieve the control precision of microns and even nanos in the servo system because the operating frequency of stator is very high and the rotor or the slider is light. Then ultrasonic motor's response is very quick and its displacement resolution is very high. (5) No electromagnetic interference: The ultrasonic motors arc different from traditional motors. They do not produce magnetic fields and will not be suffered from electromagnetic interference in the process of running. (6) Low noise: The operating frequency band of ultrasonic motors is usually more than 20kHz and beyond the scope of human hearing. In addition. Because the motor can directly drive loads. the noise from the gear box for reducing the speed is avoided. (7) Operating in extreme environmental conditions: The rational design and appropriate selection of piezoelectric and frictional materials can make ultrasonic motors operate in extreme environmental conditions (vacuum or high/low temperature) .
2. Disadvantages (1) Small power output. low efficiency: Ultrasonic motors have two energy conversion processes. The first process converts electrical energy into mechanical energy by reverse piezoelectric effect. The second process changes vibration of the stator into macro one-direction movement of the rotor by friction between the stator and rotor. Energy loss emerges from these two processes. especially in the latter. As a result. efficiency of ultrasonic motor is low. At present. the efficiency of traveling wave ultrasonic motors is about 30 % and output power is less than sow. (2) A short operational life and unsuitability of continuous operating: Friction and wear problems exist at the interfaces between the stator and rotor in the process of friction drive. In addition. high-frequency vibration can lead to fatigue damage of the rotor and piezoelectric materials especially when the power output is big and environmental temperature is high. As a result. operational life is shortened. and performance will be reduced after continuous operating for a long time. (3) Special requirements for the drive signals: In order to excite the resonance of the stator. the motors have special requirements for amplitude. frequency and phase of excitation signals. When the motor temperature changes. frequency of excitation signals for piezoelectric elements needs appropriate adjustment to maintain the stability of the output performance. Therefore. the circuit of ultrasonic motor drivers is complex.
1. 2. 2
Classification of Ultrasonic Motors
Ultrasonic motors with design flexibility and structural diversity have no uniform method of elassification. Table 1. 3 lists some classifications from different viewing angle.
Chapter 1
Table 1. 3
Introduction
9
Classification of Ultrasonic Motors
Viewing angle Wave propagation method
Type Traveling wave, Standing wave
Movement output way
Rotational, Linear
Contact state between the stator and rotor
Contact, )Jon-contact
Excitation conditions of stator by piezoelectric companets
Resonant, )Jon-resonant
Number of degree of freedom of the rotor
Single-degree of freedom, Multi-degree of freedom
Displacement of operating mode in direction
Out-plane, In-plane
Geometric shape of stators
Disk, Ring, Bar and Shell
Rotary directions
Unidirectional, Bidirectional
As ultrasonic motors arc typical products of utilizing vibrations, the classification according to the vibration type can essentially reflect the characteristics of these motors. Herewith, ultrasonic motors can be divided into five categories as follows.
1. Based on longitudinal vibration The motor based on the longitudinal vibration mode belongs to the standing wave motor. This kind of motor uses the Langevin vibrator with high converting efficiency from electrical energy to mechanical energy. However, the abrasion of the friction material on the contact surface is serious. The first standing wave ultrasonic motor proposed by Sashida bclongs to this kind of motors. The longitudinal vibration of the stator in one direction can be transformed into the rotating motion of the rotor through the deformation of the flexible sheet. In 1998, Kurosawa proposed a linear ultrasonic motor with higher drive efficiency using the composite mode of two longitudinal vibration1l6 -. Experiment proved that the large power density of 76 W / kg can be achieved, maxim urn ou tput force is 51='J and maximum speed is O. 55m/s.
2. Based on composite mode of longitudinal-bending vibration In 1989, Tomikawa designed a linear motor based on the longitudinal and bending modes of a rectangular plate[17: , as shown in Fig. 1. 12. This motor used the first longitudinal vibration mode and the fourth bending mode of rectangular plate to achieve elliptical motion of the driving feet. The efficiency of prototype by experimental measurement was 20. 8 %. Later, Tomikawa also put forward a flat linear ultrasonic motor using the first longitudinal mode and the eighth bending vibration modc: 10 ]. The experimental result was that no-load maximum speed of prototype was O. 7 m/ s, and the maximum thrust was 4N. The motor was characterized by simple structure: flat shape and fast speed, which were particularly suited for the transmission of light-thin objects such as paper, card and the like. Before and after 1995, Nikon and NEC Co., Ltd. produced the motor products of this type, rcspccti vcly.
10
Ultrasonic Motors Technologies and Ap plicalions
In 1990, Oonishi designed a IT-shape linear ultrasonic motor with biped structure: 1R ], as shown in Fig. 1. 13. Two stacked piezoelectric ceramics in tilt layout stimulated longitudinal and lateral bending modes of the leg parts of the IT-shape elastomer, which synthesizes elliptical motion for driving the guide rails. The excitation frequency was about 90kHz and the phase difference of voltage imposed was 90 degrees. This motor's no-load speed was 30cm/s, and the maximum thrust was ION. In 1993, SU)JSYN Company manufactured and commercialized this IT-shape linear ultrasonic motor for X- Y positioning systems, which became the first application of linear ultrasonic motors.
Stacked piezoceram ic
Elastomer
,- -- - -- ~
I Sy'
O. Mode
~
\C0!; >~ L I Mode
Fig. 1. 12 Tomikawa's plate type longitudinal/bending motor
Fig. 1. 13 stator
IT
shape linear motor
Cylindrical-ball type ultrasonic motor with multi degrees of freedom also uses a composite mode of the longitudinal-bending vibration of the cylindrical stator and its operating mechanism will be described in detail in Chap. 7.
3. Based on composite mode of longitudinal-torsional vibration As shown in Fig. 1. 11, Kurosawa in 1991 developed longitudinal-torsional hybrid motor-19-20J. The unique characteristic of this motor was that stacked piezoelectric vibrator produced longitudinal vibration. This vibration could possess larger amplitude in conditions of low-voltage and non-resonance. The rotor's diameter was 50mm and total length was 82mm. The motor's no-load rotational speed measured was 100r/min, the maximum torque was O. 7)J'm, and the maximum efficiency was 33% when pre-pressure of 90N was applied on the rotor, and voltage 31 V,rn, imposed on torsional vibrator. Fig. 1. 15 shows a longitudinaltorsional hybrid ultrasonic motor with a brush developed by PDLab L21 -. Within the motor longitudinal and torsional piezoelectric ceramics were placed in the stator and rotor, respectively. This design could adjust structural parameters of the stator and rotor individually to keep the modal frequencies of the longitudinal and torsional vibrations as close as possible. The rational design of the structure of the motor could increase the pre-pressure on the contact surfaces in the operating process, which could improve output performance and operating efficiency. Because the torsional vibration piezoelectric ceramics were placed in the rotor, the brush was used for supplying electric power to the rotor. The motor was called the brush type longitudinal-torsional hybrid ultrasonic motor. The motor's diameter was 45mm, length was 210mm and maximum output torque was 2. 5N'm
Chapter 1
Introduction
11
when the operating frequency was 25kHz. @ - Nut
Stacked
Shaft
piezoceramic
Torsiona1 vibrator Piezoceramic -i§J~~
Kurosawa's longitudinal-torsional motor
Fig. 1. 14
Brush type longitudinal-torsional motor developed by PDLab
Fig. 1. 15
4. Based on bending vibration According to the structure of stator, ultrasonic motors based on bending vibration mode can be divided into three categories: bar-type, ring-type, and disktype, which all belong to traveling wave ultrasonic motors. In recent years, USMs with bar-type stator based on modes of the out-of-plane bending vibration have became a hotspot in the research area of micro actuator because of their advantages of simple structure, manufacturing convenience, and low cost. Some ultrasonic motors with bar-type stator have been applied to the micro-lens focusing system and medical endoscopy system. In 1988, Kurosawa designed a bar-type traveling wave ultrasonic motor with dual rotors: 22 ] , as shown in Fig. 1. 16. The stator also used Langevin vibrator and piezoelectric ceramics divided into four areas to excite two orthogonal bending modes. This motor's diameter was 20mm, maximum output torque was O. lS)J°m, and no-load maximum speed was 300r/min. This motor was very suitable for automated production, and had been widely applied in the lens focusing system of EOS camera of Canon Co., Ltd. In 1998, Morita used the method of hydrothermal deposition in the metal surface to obtain the piezoelectric thin film and successfully developed a kind of micro high-performance bar-type ultrasonic motor[20 "] shown in Fig. 1. 17.
The
6,um-thick PZT thin films were deposited in the surface of the titanium tube, on the external surface of which four electrodes were formed. The motor started to operate when alternating voltages with a phase difference of 71:/2 were imposed on electrodes and the middle part of the stator was connected to earth. Based on the d 31 effect, two pairs of the piezoelectric components in opposite position excite the corresponding bending modes of metal cylinder. The rotor's diameter was 2. 1mm, length was 10mm, maximum rotational speed was 880r/min under the excitation voltage 15Vpp
,
and the maximum torque was 7. 6,u)Jom. Two-axis me-
chanical arm driven by this motor could achieve step movement and lift an object with weight 109. This simple novel structure of the motor opened a new way for the development of bar-type micro USM['s:.
Ultrasonic Motors Technologies and Ap plicalions
12
Fig. 1. 16
Kurosawa's bar-type motor
Micro USM based on piezoelectric film
Fig. 1. 17
5. Based on in-plane vibration In-plane vibration has three types: extension-eontraetion, bending, and torsion. In 1989, Takano used in-plane extension-contraction and bending vibration mode for developing an ultrasonic motor, as shown in Fig. 1. 18 L26 -. When alternating voltage signals with phase difference rr/2 were respectively imposed on circular piezoelectric ceramics with two areas, radial and tangential movement of drive point A(A') synthesized the elliptical motion, which drove the rotor. The rotor's diameter was 10mm, thickness was 2mm, operating frequency was 13. 3kHz, maximum output torque was 40mN 'm, and efficiency was 3. 5 %. In 2008, Tieying Zhou developed a linear ultrasonic motor, which used the inplane extension-contraction vibration modes of a hollow cylinder L27 -, as shown in Fig. 1. 19.
Pre-pressure
Fig. 1. 18
Pre-pressure
USM bascd on in-plane modes
Linear USM based on in-plane extension-contraction mode
Fig. 1. 19
In addition, there are some ultrasonic motors based on other modes such as a composite mode of the torsional and bending vibrations[2R:, a composite mode of the longitudinal and shear vibrations L29 -.
1.3
Comparison with Electromagnetic Motors
As mentioned above, ultrasonic motors are new concept motors that can directly drive loads devices and its operating principle is totally different from the electromagnetic motor. In the following sections we will compare the ultrasonic motor with the traditional DC motor in three aspects.
Chapter 1
1. 3. 1
13
Introduction
Load Characteristics
Figure 1. 20 shows the comparison of the load characteristics (measured) of DC motor with USM. It can be seen that when DC motor approaches the no-load speed. its efficiency is maximum and output torque is smaller. On the contrary. the ultrasonic motor's efficiency is maximum in conditions of the lower speed and higher torque. Therefore, USM is suitable for operating at low speed and high torque. and can directly drive loads. PowerfW
Efficiency!"!. Speedl(r/m in)
10
100
8
80
6
60
4
40
2
20
0
0
Maximum efficiency at high speed and low torque / power
20000
15000
10 000
5000
0
0
14
21
28
35
42
49
Out]JlIl torque !(N·m) (a) DC Molor PowerfW
Efficiency!"!. Speedl(r/min)
8.0
100
200
6.4
80
160
4.8
60
120
3.2
40
80
1.6
20
40
0
0
0
Maximum efficiency at low speed and high lorque
0
0.24
0.48
072
0.96
1.20
Output lorque I(N 'm) (b) Ullrasonic motor
Fig. 1. 20
1. 3. 2
Comparison of the measured load characteristics of DC motor with USM's
Energy Transform of Motors and Their Micromation
Electromagnetic motors convert the electromagnetic energy into mechanical energy based on the electromagnetic principle. The magnetic field is produced by the current applied to energize winding on a stator or a permanent magnet stator and there are many coils around the rotor. When the current is applied to drive coils
14
Ultrasonic Motors Technologies and Ap plicalions
on a stator, the magnetic field drives a rotor to rotate. Therefore, the electromagnetic motor often consists of the stator and rotor, and a gap exists between them, which do not contact each other. Ultrasonic motors use the inverse piezoelectric effect of piezoelectric material to achieve the conversion of electrical energy to mechanical energy, and then ultrasonic vibrations of the stator arc transformed into the macro one-direction movement of the rotor through the friction between the stator and rotor. Except non-contact type ultrasonic motor (Chap. 11), the stator and rotor are in contact. Figure 1. 21 shows the comparison of efficiency vs. size of electromagnetic motors with ultrasonic motors:]:. It can be seen that the efficiency of an electromagnetic motor sharply deelines as the motor's diameter decreases to below 10 mm, while the efficiency of ultrasonic motor smoothly changes. At the same time, electromagnetic motor is more difficult to achieve miniaturization because its rotor must be rounded with coils and its structure is complex. As a contrast, the structure design of ultrasonic motor is more simple. Particularly bar-type traveling wave ultrasonic motor developed since 1990s is very suitable for miniaturization because of its simple structure, machining convenience, and low cost. New Scale Technology, Inc. uses a bar-type ultrasonic motor for the auto-focus system of a mobile phone camera. The motor's cross section is 1. 5mm XL 5mm, operating trip is 30mm, maximum speed is 8mm/ s and maximum output force is 1:'\1. In addition, Konica-Minolta , ]ohnson-:'\Janomotion, and Sam sung Electro Mechanics arc all using ultrasonic micro-motor for the camera lens.
1. 3. 3
Transient Response Characteristics
A comparison of transient response characteristics of the ultrasonic actuators with electromagnetic actuators' in welding system c,,: is shown in Fig. 1. 22. Obviously, the ultrasonic actuator responds much faster than the electromagnetic actuators. From zero speed to the stable speed, it only costs several milliseconds, and the stopping time to zero speed is even shorter. For the ultrasonic and electromagnetic motors , there are the similar rules. See Chap. 11 in detail. 40.---------------------, _... ->l<.
">.u
., to
"u i::
~
30 20
250
= ~u ~
<>
UJ
10
,, ,,
'" ~
Ci
- - - - " Electromagnetic motor - - Ultrasonic motor
OL---~----~----~--~
o
5
10
15
20
Relationship
and size for and USM
of
t ,,
I
I
150
I j
(
II I
100
,
•
1'\
"
I
I I
I
50 00
I
" ---- Electromagnetic actua tor ",,/ - - Ullrasonic actuator
0.01
0.D2
0.03
0.04
0.05
Re ponse timeJs
Motor diameter/mm
Fig. 1. 21
..
200
efficiency
electromagnetic
motor
Fig. 1. 22 Comparison of start-up characteristics between electromagnetic and ultrasonic actuator
Chapter 1
1.4 1. 4. 1
Introduction
15
Applications and Development Trends of Ultrasonic Motors Applications
As mentioned above. ultrasonic motors have some unique advantages. So it promises a broad application prospects in many fields such as micro-robot. automobile, aerospace, precision positioning system, optical instrument, and medical endoscope. Color Fig. 1 lists some successful application examples in the world. Ultrasonic motors show strong vitality and board market potential sincc they are invented. Starting from the 1980's. many Japanese scientists have dedicated to commercializc the prototypc motor developcd by Amcrican or Soviet rescarchers. At present, many Japanese famous universities and companies are researching and producing ultrasonic motors. Japan now holds many invention patcnts of ultrasonic motors in the world. Canon Co., Ltd. has established a special production line of ultrasonic motors. USA. Germany. France, Britain, etc. have been putting in a lot of manpower and resources to develop ultrasonic motorsL31-32J. Applications of ultrasonic motors are gradually promoted in China. As shown in Figs. 1. 23 and 1. 21, PDLab successfully uses ultrasonic motors in the injector of MRI systcm and two-dimcnsional wing flutter modcl. Some other applications are introduced in Chap. 16 in details.
Fig. 1. 23 USM used in the injector of MRI system
1. 4. 2
Fig. 1. 24
USM used in airfoil flutter
model
Development Trends
As several types of linear ultrasonic motors and the ultrasonic motors with multi degrees of freedom have been developed in recent years, the development trend of USMs can bc summarizcd as follows:
1. Development of new frictional and piezoelectric materials for improving ultrasonic motors' adaptability to the extreme environment. Because ultrasonic motors transfer torque through the friction coupling. thc abrasion and fatigue problem of friction material on the contacting surface is incvitablc, which greatly limits applications of the motors. At prcsent, ultrasonic
16
Ultrasonic Motors Technologies and Ap plicalions
motors are only applied to the intermittent operating occasions, for example, total operational time of the ultrasonic motor in camera focus system is less than 15 hours, total operational time in car window switches or scat adjusting devices is less than 500 hours. In the last two years, Canon Co., Ltd. has used traveling wave ultrasonic motor for color copiers and the required operational life of the motor is more than 3 000 hours. Some applications also request longer life. To that end, many researchers are trying to find the new friction materials in order to increase the life of ultrasonic motors. Take the ultrasonic motors of Shinsei Corporation as an example: the biggest improvement for the past 10 years could be the friction materials including the material formula, manufacture, and adhesive bonding techniques. In order to use ultrasonic motors for aerospace engineering, the research on motors' adaptability to the environmental conditions is necessary. To complex environment of the space, National Aeronautics and Space Administration (NASA) and jet Propulsion Laboratory (jPL) have done a lot of research on the operating motors in the high/low temperature and vacuum environment C03 :. They applied USR-30 ultrasonic motors of Shinsei Corporation for a destructive testing of 67-hour operating in Cryovae conditions of - 80°C and 25mtorr (1 torr = 1. 33322 X 10 2 Pa). Based on testing results, taking a number of special measures, :'\JASA and jPL developed their SRPD type USM, which experienced the operating of 336 hours (65 hours in the conditions of -80°C and 25mtorr, 271 hours in the ones of -150°C and 16mtorr) and exhibited a good Cryovac characteristics. The researches show that in order to achieve good low temperature properties, in addition to improve adhesive materials and adhesive bonding techniques, it is necessary to improve the low temperature performance of frictional and piezoelectric materials and develop some new materials suitable for low-temperature environment. When the temperature drops to -10°C, the performance of series of PZT piezoelectric ceramics may decrease because of the increase of hysteresis loss. Piezoelectric ceramics remains only 25 % capability at the temperature of - 210°C. However, The piezoelectric properties of single-crystal relaxor ferroelectric piezoelectric ceramics (1 - .z:) Pb( Mg1/O Nb2/3 ) 0, - .z:PbTiO, (solid solutions of lead magnesium niobate-lead titanate, abbreviations PMN-PT) and (1.z:)Pb(ZnJ/3:'\Jb'!3)03 -.z:PbTi03 (solid solutions of lead zincum niobate-lead titanate, abbreviations PZN-PT) in the temperature - 240°C arc still better than the ones of piezoelectric ceramics in 30°C L31 . • Obviously, PMN-PT single crystal will effectively take the place of the traditional piezoelectric ceramic materials.
2. Miniaturization and integration of ultrasonic motors As mentioned above, the ultrasonic motor has no coil and its structure is simple, so it is easy for manufacture and it possesses higher energy density compared with the traditional motor. From Fig 1. 21, we can sec the efficiency of USMs is nearly independent of their sizes, which makes USMs extremly useful as actuators of micro-electro-mechanical system (MEMS). Therefore, miniaturization and integration are important development trends of ultrasonic motors. Though there is another small and special electrostatic motor, which can be
Chapter 1
Introduction
17
manufactured based on the Ie processing and can also be integrated together with the driving circuit, but its output torque is very small because of the limit of operating principle. The energy density of electrostatic motor is Eo E' /2. where Eo denotes the air dielectric constant and E denotes electric field intensity. When the size of the gap is IfLm, E::::::::IOR(V/m). The energy density of ultrasonic motors using the reverse piezoelectric effect of PZT can also be calculated by EE' /2, where E denotes the dielectric constant of piezoelectric ceramics and E is I 300Eo. Therefore, it can be said that ultrasonic motors have higher energy density[os:. For example, output torque of 5. 5mm in diameter electrostatic motor presented by Ref. [36Jwas only 25n)J'm, which was difficult to meet the needs of practical engineering applications. Miniature linear ultrasonic motor presented by Yoon is shown in Fig. 1. 25: shaft diameter is Imm. length is 6mm, its output force is 160mN, and velocity is 12mm/ s. Miniature bar-type ultrasonic motor presented by Ref. [37J is shown in Fig. 1. 26: diameter is 2. 1mm, length is 10mm, rotational speed is 570r/min, output torque is 1. 8mN'm, and efficiency is 25%.
Fig. 1. 25 Micro linear USM developed by Yo on
Micro bar-type traveling wave USM developed by Koe
Fig. 1. 26
3. Combination of piezoelectric actuator and biomedical engineering In modern biomedical engineering, it is indispensable to manipulate cells such as processing. transfer, separation, and fusion. or cellular materials (karyons. chromosomes, genes) transfers, restructuring, stretch, and fixation. For cells of micro size. a key technique is the precise positioning one and i t always requires the resolution of dozens of nanometers. Driving devices with high positioning accuracy and fine resolution are needed to accomplish this kind of manipulation. At present, these operations are manually implemented by professionally trained technical personnel with low efficiency and low success rate. USMs with high positioning resolution and fast response can successfully solve the problem. It can be deduced from the fact that Japan developed three-dimensional micro system which operates on the leukocyte, its positioning resolution is o. IfLm and operating range is 582fLm X 582fLm X 52fLm because human leukocyte diameter is about IOfLm. This system used laminated piezoelectric ceramics as actuator, had two-finger micro manipulator and could imitate the movement of chopsticks[08:.
Ultrasonic Motors Technologies and Ap plicalions
18
This system can also be used for surgery operating to manipulate the glass ball of diameter 2f1-m. The adoption of precision driving technique can improve efficieney, simplify the operating and realize the automation of the bioengineering. In the laboratory, Japanese scholars recently developed a set of automated cellpuncture operating system based on the nano-positioning technique of an inertia type linear ultrasonic motor and image processing technique, as shown in Fig. 1. 27: 39J • Using the modern micro-fabrication technique, the concept of drug delivery is proposed and can greatly improve the traditional delivery method for oral peptide and oral protein. Fig. 1. 28 shows Drug Delivery System based on the linear ultrasonic motor developed by New Scale Technology, Inc. in USA.
USM used in cell-puncture micro operating system
Fig. 1. 27
Fig. 1. 28
USM used in drug delivery system
In particular, it should be pointed out that the surface acoustic wave ultrasonic motor developed by Kurosawa possesses smaller loss, higher efficiency, and smaller volume compared with the traveling wave ultrasonic motor. At present, the surface acoustic wave motor with dimensions of 4mmX 4mmX 3mm has been developed, and its operating frequency range is lO-lOOMHz. When it is used as a step motor, O. Snm stepping motions and every step's response time of O. 2ms can be achieved: 40J • This ultrasonic motor with high spatial resolution has shown broad application potential in the fields of computer science and biomedical engineenng.
References [ 1]
[ 2] [ 3]
J Wallasehek. Ultrasonic motor research in Germany-past, present, future. Proceedings of the First International Workshop on Ultrasonic Motors and Actuators. Yokohama, Japan: Tokyo Institute of Technology, 2005. W Willams, W Brown. Piezoelectric motor. US Patent, 2439499, 1942-08-20. F J Britten. Britten's Watch and Clock Maker's Handbook: Dictionary and Guide. )lew York:
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Areo Publishing Co., 1978: 109. V V Lavrinenko, M )lekrasov. Piezoelectric motor. Soviet Patent, 217509, 1965. H V Barth. Ultrasonic drive motor. IBM Technical Disclosure Bulletin, 1973,16(7): 2263. V Vishncvsky, V Kavertsev, I Kartashev, ct al. Piezoelectric motor structures. US Patent, 4019073, 1975-08-12.
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P Vasiliev, R Klimavichjus, A Kondratiev, et al. Vibration motor control. UK Patent, GB
Chapter 1
[ 8J [9 J [lOJ [l1J [l2J [l3J
[14J [15J [l6J
Introduction
19
2020857A, 1979-11-21. T Sashida. Trial construction and opcration of an ultrasonic vibration drivcn motor. Oyo Butsiuri, 1982, 51(6): 713-718. T Sashida. Motor dcvicc utilizing ultrasonic oscillation. US Patent, 1562371, 1981-05-16. Takashi Kenjyo, :'-Icnsei Sashida. Introduction of Ultrasonic Motor. Japan: Sougou-Dcnshi Publishcr, 1991. (in Japanese) A Kumada. A piezoelectric ultrasonic motor. Japanese] ournal of Applied Physics, Supplement, 1985, 24(2): 739-741. Isc Yukihiko. Ultrasonic motor. Journal of the Acoustical Society of] apan, 1987, 13 (3): 184-188. K Uehino. Piezoelcetrie actuators/ultrasonic motors-thcir dcvelopments and markcts. IEEE International Symposium on Applications of Ferroelectrics. PA, USA: University Park, 1994: 319-324. J Wallaschek. Piezoelectric ultrasonic motors. Journal of Intelligence Material Systems and Structure, 1995, 6(1): 71-83. Tieying Zhou, Shuxiang Dong. Circular ultrasonic vibrator and the micro-motor driven by this vibrator. Chinese Invention Patent, ZL89109320, 1989-12-21. (in Chines c) M K Kurosawa, 0 Kodaira, Y Tsuehitoi, ct at. Transducer for high speed and large thrust ultrasonic linear motor using two sandwich-type vibrators. IEEE Transactions on Ultrasun-
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[22J
Y Tomikawa, T Ogasawara, T Takano. Ultrasonic motors-constructions /characteristics/ applications. Ferroelectrics, 1989, 91:163-178. Kazumasa Onishi. Principle and mcchanism of ultrasonic linear actuator. Labor-saving and Robotization, 1990, 112: 165-170. (inJapanesc) M Kurosawa, S Ueha. Hybrid transducer type ultrasonic motor. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 1991, 38(2): 89-92. S Ucha, Y Tomikawa. Ultrasonic Motors-Theory and Applications. USA: Oxford University Press, 1994. Zhcng Tao. Study on Hybrid Ultrasonic Motor Using Longitudinal and Torsional Vibration Modes. Dissertation for thc Dcgrec of Doctor of Philosophy. Nanjing: :'-Ianjing Univcrsity of Aeronautics and Astronautics, 2006. (in Chincsc) M Kurosawa, K Nakamura, T Okamoto. An ultrasonic motor using bcnding vibrations of a short cylinder.
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1989, 36(5): 517-521. T Morita, M K Kurosawa, T Higuchi. A cylindrical micro-ultrasonic motor using PZT thin film depositcd by single process hydrothermal mcthod (Diametcr 2.1mm, L10mm Stator IEEE Transactions un [lltrasonics, Ferruelectric'i and Frequency Cuntrol,
1998,45(5): 1178-1187. T Morita, M K Kurosawa, T Higuchi. A cylindrical shaped micro ultrasonic motor utilizing PZT thin film (Diameter 1. 4mm and L5. Omm stator transducer). Sensors and Actuators A: Physical, 2000, 83 (3): 225-230. Hua Zhu, Chunsheng Zhao. The review of rotational ultrasonic micro-motor. Piezoelectrics & Acuustooptics, 2005, 27(6): 627-630,642. (in Chinese) T Sashida, T Kenjo. An Introduction to Ultrasonic Motors. Oxford: Clarendon Press, 1993. Tieying Zhou. An integrated optical auto-focus system driven by a nut-type USM. 5 th International Workshop on Piezoelectric Materials and Applications in Actuators. PA, USA:
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Pann. State University, 2008. A Kumada. A piezoelectric ultrasonic motor. Japanese] ournal of Applied Physics, Supplement, 1985, 24(2): 739-741. P Bouchilloux, B Koc, K Uchino. New concept for resonant longitudinal-shear ultrasonic motor. Materials for Srnart Systerns, Syrnpusiurn (Materials Research Society Pruceedings) ,
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A Henke, M A Kummel, J Wallaschck. A piezoelectrically drivcn wire fceding systcm for
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Chunshcng Zhao. Ultrasonic motor tcchniqucs for 21" ccntury. Engineering Science, 2002,
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S S Lih, B C Yoscph, W Grandia. Rotary ultrasonic motors actuatcd by traveling flcxural
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waves. SPIE, 2004, 3041: 912-917. Physics Letters 86, 053501 (2005). nal of Microelectromechanical System, 1992, l( 1): 44-5l.
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B Koc, S Cagatay, K Uchino. A piezoelectric motor using two orthogonal bending modes of a hollow cylindcr. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 2002, 1(19): 195-500. T Tanikawa, T Arai. Development of a micro-manipulation system having a two-fingered micro-hand. IEEE Trans. Robot. Autom. , 1999, 15: 152-162. T Higuchi. Automatic micro manipulation systcm for ccll manipulation. [2007-05-23]. http: / /www.intellcct.pc.u-tokyo.ac.jp/rcscarch/manipulator/manipulator_c.htm!. T Shigematsu, M Kurosawa, K Asai. Sub-nanometer stepping drive of surface acoustic ultrasonic motor. IEEE Int. Conf. Nanotechnology. San Francisco, CA, 2003, 2: 299-302.
Chapter 2
Fundamentals of Piezoelectricity and Piezoelectric Materials for Ultrasonic Motors The development of ultrasonic motors is a highlight in the application of piezoelectric materials. Ultrasonic motors take advantage of converse piezoelectricity of piezoelectric materials to yield mechanical output from converting electrical energy. It is no doubt that piezoelectric materials are in the central position that controls thc pcrformancc of thc dcviccs. In this chaptcr, wc will rcvicw piczoelcctric materials and their properties from the aspect of their applications in ultrasonic motors. The chapter begins with a brief overview of historical development of piczoelcctric matcrials, and thcn a dctailcd cxplanation of thc elcctrical and mcchanical properties of piezoelectric materials will be discussed in the second and thc third scctions, whcrc thc piczoelcctric constitutivc cquations arc spccially strcsscd. Piczoelcctric vibrators, as basic units of piczoelcctric dcviccs, and thcir vibration types will be described in sequence. In the last two sections, the application of piczoelcctric matcrials in ultrasonic motors and somc advanccs in novel materials will be introduced.
2. 1
2. 1. 1
Development and Classification of Piezoelectric Materials Historical Development of Piezoelectric Materials
In 1880, Pierre Curie and] acques Curie reported a number of crystals such as quartz, topaz, tourmalinc and Rochcllc salt that could display surfacc chargcs when they were mechanically stressed. This phenomenon that materials produce elcctricity in rcsponsc to applicd strcss is defincd as thc dircct piczoelcctric cffcct, of which thc discovcry is thcreforc crcditcd to thc Curic brothcrs. Thc inverse process of the piezoelectric effect, that is say, an external electric field applicd to piczoelcctric matcrial gcncratcs deformation in thc matcrial, is defincd as the converse piezoelectric effect- 1- 6J • Frcnch scicntist Langcvin thcn dcvelopcd thc first scrious application for piczoelectric materials in 1916 by using quartz crystals to build transducers for submarinc dctccting. Thc discovcry of BaTi03 piczoelcctric ccramic in World War II was rcgardcd as a milcstonc of thc dcvelopmcnt of piczoelcctric matcrials. In contrast with piczoelcctric singlc crystals, BaTi03 ccramic is rathcr casy to prcparc, and can bc form cd into spccific shapcs and pol cd in arbitrary dircctions.
C. Zhao, Ultrasonic Motors © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011
22
Ultrasonic Motors Technologies and Ap plicalions
These advantages brought BaTi03 to be rapidly applied in ultrasonic transducers, high-frequency transducers, pressure sensors and other piezoelectric devices. It has gone through three stages in discovering and understanding the piezoelectricity of ceramics: the first stage was the discovery of high dielectric constant in the ceramics; then at the second stage it was realized that the high dielectric constant was originated from the ferro electricity of the materials; poling processing for the ceramics was finally established at the third stage. Lead zireonate-lead titanate (PZT) system is another progress in the field of piezoelectric materials. In 1951, ] affe et al from the United States reported much better piezoelectric properties found in PZT solid solution ceramics and proposed the concept of morphotropic phase boundary (MPB)L1J. Based on this concept series of the currently most widely used piezoelectric ceramics were developed. Recent research focus of piezoelectric materials is on the relaxor ferroelectrics of Pb (Mg x Nb j - x ) 0 3 -PbTi0 3 (PMN-PT) and Pb(ZnE:'\Jbl-E)03-PbTiO, (PZ:'\J-PT) single crystals, which were reported with outstanding electromechanical coupling properties L5 -
2. 1. 2
Classification of Piezoelectric Materials
The currently used piezoelectric materials cover three classes: CD Inorganic piezoelectric materials, including piezoelectric monocrystalline materials and piezoelectric ceramics which consist of massive fine crystals. Piezoelectric ceramics present advantages such as strong piezoelectricity, high dielectric constant and can be easily formed into various shapes. But they arc usually with low mechanical quality factor, large electric loss and poor stability. These characteristics make piezoelectric ceramics suitable for high-power transducers, wide-band filters and so on. Piezoelectric single crystals provide high mechanical quality factor and excellent stability but low piezoelectric coefficient and low dielectric constant, and their shapes for devices arc restricted because of the difficulty in machining these crystals. Piezoelectric single crystals can be used in devices such as vibrators to control standard frequencies, high-selectivity filters (usually with high frequency and narrow-band), high-temperature ultrasonic transducers and so on. CZ)Organic piezoelectric materials, also known as piezoelectric polymers, e. g., polyvinylidene fluoride (PVDF). ] ust like other polymers, piezoelectric polymers possess excellent flexibility, low density, small impedance, as well as reasonable piezoelectric coefficient. Piezoelectric polymers have been rapidly applied in devices for underwater ultrasonic measuring, pressure sensing, and explosion igniting. However, the low piezoelectric stain constant of piezoelectric polymers has restricted their applications as active transducers. Gil Piezoelectric composites, in which piezoelectric ceramics and polymers are incorporated together. As a result, the piezoelectric properties of the composites are enhanced comparing with their initial components. Furthermore, the composites may present novel properties that do not exist in these single components. Piezoelectric composites can ha ve large piezoelectricity, strong strength and low density, and their outstanding machinability makes them easy to be fabricated into large area films or other complicated forms. Nowadays, piezoelectric composites have already been widely
Chapter 2
Fundamentals of Piezoelectricity and Piezoelectric'"
23
used in hydroaeoustie. eleetroacoustic. ultrasonic and medical applications.
Electrical Properties of Piezoelectric Materials
2.2
This section is concerned with some important electrical properties of piezoelectric materials. Parameters such as dielectric constant. dielectric loss. electrical quality factor. ferroelectric polarization and Curie temperature will be explained in detail, with which the dielectric and ferroelectric properties of piezoelectric materials are described.
2. 2. 1
Dielectric Properties and Dielectric Loss
1. Dielectric polarization and dielectric constant[l, :1, 7J All piezoelectric materials are dielectrics. Response of dielectric materials to external electric field will produce net dipole moments or produce changes in dipole moments along the electric field direction. This process is called the polarization of dielectrics. The induced and orientational polarization in external electric field is responsible for the mechanism of polarization. As a result of polarization. electric charges appear on the outside surfaces of the dielectrics. The quantity P is the basic parameter describing the polarization of the dielectric and denotes the dipole moment per unit volume. Under moderate external field. the polarization P is proportional to the electric field strength and can be expressed by the linear equation
P where
Eo =
=
(2. 1)
EoXE
8.85 X 10- 12 F/m is the permittivity (or dielectric constant) of free
space and the dimensionless constant X is called the susceptibility of the medium. which is a second-rank tensor. When P and E are collinear. and X is simply a scalar. the electric displacement vector D can be written as D
If we define
Eo E, =
E.
=
Eo E
+P
=
Eo
(1
+ X) E
=
EoE,E
(2. 2)
then (2. 3)
where E referring to the dielectric constant of the material. is an important parameter used to describe the dielectric property of the material in static electric fields. In isotropic materials. E is simplified as a scalar. In anisotropic materials such as single crystals. P. D. and E are vectors. precisely. are first-rank tensors. The coefficients E and X connecting them are then second-rank tensors. The number of the independent components of a dielectric constant tensor depends on the symmetry of the crystal structure. Materials with higher crystallographic symmetry always have fewer independent components in the dielectric
24
Ultrasonic Motors Technologies and Ap plicalions
constant matrix. For example, triclinic crystal, which is with the poorest symmetry, has six independent components of £jj, £22 , £33 , £j2 , £j3 , and £23' On the other hand, cubic crystal has the highest symmetry so that only one independent component exists. Unpolarizcd polycrystallinc ceramic is isotropic and presents identical dielectric constant in all directions. However, the poled piezoelectric ceramic presents anisotropy because of the existence of remnant polarization along the poling direction. The dielectric constant component in this direction is different from these of the other two directions. In practice, the expression of dielectric constant for hexagonal crystal system can be taken to describe the poled piezoelectric ceramic, in which two independent components £jj =£22 and £33 arc used. Taking into account different boundary conditions during test, dielectric constants can be sorted into the stress-free dielectric constant and the mechanically clamped dielectric constant, symbolized as £;j , £i; and £fj , d3' respectively.
2. Dielectric loss and electrical quality jactor[l, 3, 6, 8J When an electric field applied to dielectrics, energy loss always occurs because of polarization relaxation or leakage and other reasons. This energy loss in dielectrics is defined as the dielectric loss. If a dielectric is suddenly exposed to an electrostatic field, the polarization building up from zero to the final value is not instantaneous but takes a finite time. This phenomenon is described by dielectric relaxation and the finite time is defined as relaxation time. In a high frequency AC field, the response of the dielectric may not follow the change of the field, the polarization therefore lags behind the field and leads to dielectric loss. The polarization relaxation will also result in a difference between the dynamic dielectric constant and the static dielectric constant in the material. Part of the energy of dielectric loss is consumed to rotate the dipole moments because of polarization lag. The energy finally transforms into heat energy and dissipates. Current leakage in dielectric is another factor that causes dielectric loss, especially at high temperature or in strong electric field strength. The energy also dissipates in the form of thermal effect. In ideal dielectrics without dielectric loss, the phase of the current inside leads 90° ahead of the phase of the voltage. However, in actual piezoelectric materials, the energy loss makes this leading angle rp less than 90 degrees. The complementary angle G of the angle rp is then defined as the loss angle. Tangent of the loss angle tanG is defined as the quotient between the active power P and the reactive power Q in a dielectric. One can image that there is a resistance R in the dielectric to consume part of energy. The current in the dielectric is then divided into two parts: IR goes through the resistance to cause energy loss; Ie passes the capacitor without losing. The tangent of the loss angle of dielectrics can be written as tanG
=
IR Ie
1
=
wCR
(2. 1)
In which w stands for the angular frequency of the alternating electric field, and C is the static capacitance of the dielectric with electrodes.
Chapter 2
Fundamentals of Piezoelectricity and Piezoelectric'"
25
Since tanB is proportional to the energy loss, naturally it is used to characterize thc dielcctric loss of matcrials. In gcneral, thc more is thc dielectric loss, thc worsc is thc pcrformancc of the material. Dielectric loss is also scnsitive to thc temperature, the strength and the frequency of the electric field. Thc rcciprocal of thc tangcnt of loss anglc is defined as the electrical quality factor Q" which is shown as Q,
2. 2. 2
1 tanB
---(JJ
CR
(2. 5)
Ferroelectric Properties and Polarization
1. Ferroelectricityrl.
3, 781
The function parts uscd in ultrasonic motors and other actuators actually arc fcrroelcctric matcrials. As wc know from solid statc physics, thc structural symmetries of crystalline solids are described by 32 point groups. Dielectrics can be belonged to any of these groups. Of the 32 point groups, there are 21 non-centrosymmetric point groups. Among them, 20 point groups exhibit piezoelectric effcct as the polarization of thcse crystals varics with elastic strain. Furthermore, 10 point groups in thc crystal structurcs of piezoelectrics prcsent only onc polar axis. Crystals with the structures falling into this 10 point groups can present spontaneous polarization even without external electric field. Bcsides stress, tcmperaturc gradicnt may also inducc elcctric polarization variation in thesc crystals. The generation of polarization by temperature gradient is referred as the pyroelectricity. Among pyroelectric materials, some materials possess spontaneous polarization and the spontaneous polarization is switchable by external electric field. During switch the polarization P and thc external elcctric filed E prescnt a hysteresis loop. This property of materials is referred as the ferroelectricity. A ferroelectric crystal consists of many fine regions defined as domains, where thc spontaneous polarizations arc aligned in samc directions.
Thc boundary
bctwccn domains is callcd thc domain wall. The polarization P of a ferroelcctric prescnts a nonlinear characteristic under a strong altcrnating elcctric field and displays a ferroelectric hysteresis loop with the variation of electric field E, as shown in Fig. 2. 1, in which P is a doublc-valued function of E . Thesc behaviors makc fcrroelcctrics bc analogous to fcrromagnet in many physical propertics. The ferroelectric phase is limited to below the Curie temperature Te' Above T"
the spontaneous polarization disappcars and thc fcrroelcctric phasc turns to
paraelectric phase. Thc electric pcrmittivity of fcrroelectrics dcmonstratcs quitc complex temperature dependence. At the Curie temperature, the permittivity of the material reaches its maximum and then decreases with the increase of temperature, which can be described by the Curie-Weiss law E=
(2. 6)
Ultrasonic Motors Technologies and Ap plicalions
26
p
H
C
----------~~L--+~------~E
H
Fig. 2. 1
Ferroelectric hysteresis loop of ferroelectric materials
2. Poling of piezoelectric ceramic/:J· 9-10Many single crystals can naturally present piezoelectricity. However, as their crystal grains are randomly oriented, the sintered ferroelectric ceramics have to be polarized before they can exhibit piezoelectricity. The fine grains in the ceramics may be single-domain or multi-domain structures, the spontaneous polarizations of the domains distribute randomly and compensate each other so that no net electric dipole moments are shown. If the strength of the external electric filed is larger than the coercivity of the material, the spontaneous polarizations of the domains will be switched as much as possible to the electric filed direction. The polarized ceramics will keep a remanent polarization even the electric field is removed. Ceramics with macro electric polarizations then present piezoelectric behavior. Fig. 2. 2 displays the evolution of domain structures of ferroelectric ceramics during poling process. Before polarization, domains with spontaneous polarizations point all possible directions and the net polarization of the ceramics is zero, as shown in Fig. 2. 2(a). After polarization, the ceramics present net polarization at zero electric field, which is the remanent polarization P, of the ferroelectric ceramics, as shown in Fig. 2. 2 (c).
~
1
I
~
P,
Remnant prolongation caused prolongation
(a) Domains oriented randomly prior to polarization
Fig. 2. 2
(b) Domains rearrange along electric fie ld direction during poling proce
(c) Remnant polarization remai ns afte r remQval of DC field
Domain structures of piezoelectric ceramics before and after poling process
Chapter 2
2. 3
Fundamentals of Piezoelectricity and Piezoelectric'"
27
Properties Constitutive Equations of Piezoelectric Materials
2. 3. 1
Elastic Properties and Their Coefficients- 7 , 910_
In general, an object will make two behaviors in response to an external force. One is the position movement, including translation and rotation. The other is the deformation of volume or shape, which includes elastic deformation and plastic deformation. Considering the deformation is small enough during the performance of piezoelectric materials, we can deal with it as the elastic deformation. The elastic state of piezoelectric materials is described by stress (T) and strain(S). Fig. 2. 3 depicts the interior stress of a unit volume of piezoeleetries. As it can be seen in the figure, the stress relics on the orientation of the force as well as the plane where the force acts. The stress T can be described by a second-rank stress tensor which consists of 9 components. These components are indexed by notations with double subscript, as shown in Fig. 2. 3. Sometimes, notations of single subscript arc also used. The conversions between the indexes of double subscript and single subscript arc listed on Table 2. 1.
, 7'," 'j:,, 7'--", --- -- --:
FI_ i~ J.--------i
7~,
~,"O -
"I-- -.. y
// 0
x
x
Fig. 2. 3 Table 2. 1
72 ,J-- -.. y
Indexes of stress components in a piezoelectric bulk
Conversions between indexes with double subscript and single subscript
Double subscript
Double subscript
.II
11
Single subscript
yy
22
2
zz
33
yz
Z.T
y.T
zy
xz
xy
23
31
12
32
13
21
1
From Table 2. 1, we have
TXE = I'll = 1'1' Tyy = 1'22 = 1'2' 1'= = 1'" = 1', , Tyx = Txy = 1'12 = 1'6 , 1'= = 1'= = 1'13 = 1'5' l'yz = Toy = 1'23 = 1', Of the stress tensor components, only six ones are independent. These independent components usually are expressed by a matrix.
28
Ultrasonic Motors Technologies and Ap plicalions (2. 7)
Similarly we have six independent components in the strain tensor, expressed by the following matrix (2. 8)
In linear elastic range, the relationship between stress and strain is described by the Hooke's law that the strain component is a linear function of the stress components 6
Si =
~Si]T]
1,2,···,6 )
(2. 9)
(i=1,2,···,6)
(2. 10)
(i
=
]-1
By a linear transformation, we also have 6
Ti =
~Ci]Sj ;'=1
where Si] is the flexibility coefficient and Si]=S]i; c i] is the stiffness coefficient and we also have C i] = C]i. It is not hard to find that these elements in the matrix of flexibility coefficients and the matrix of stiffness coefficients have a relation: [Si] J=[c i]
J- 1 •
The symmetry of poled piezoelectric ceramics is approximate to those of hexagonal crystal system. The elastic constant matrix of piezoceramics is then similar to that of crystals with structural symmetry of 6mm point group. Five independent flexibility coefficients ( Sl1 , SI2 , SB , S" , s,,) are left in the matrix. The flexibility matrix of piezoelectric ceramics is as follows S12
S13
0
0
0
Sl2
S11
S12
0
0
0
S13
S13
S33
0
0
0
0
0
0
S44-
0
0
0
0
0
0
S44-
0
0
0
0
0
0
2(Sll-SI')
Sl1
s=
(2. 11)
Similarly, there are five independent stiffness coefficients in the stiffness matrix of piezoceramics, C ll , C 12 , C 13 ' C 33 ' c11 • The stiffness matrix is similar to Eq. (2. 11) in form but replacing Sij with Cij and replacing 2 with 1/2.
2. 3. 2
Piezoelectric Properties and Piezoelectric Equations
1. Piezoelectricity The ability of materials to develop or vary electric polarization when they are mechanically stressed has been known as piezoelectricity. When a piezoelectric is strained with external stress, charges displace from their equilibrium position to both surfaces, causing bound charges on the surfaces of the material. The produced charge density is proportional to the stress. This effect is the direct piezoelectricity and its mechanism is shown in Fig. 2. 4.
Chapter 2
Fundamentals of Piezoelectricity and Piezoelectric···
29
The mechanism of the converse piezoelectricity is shown in Fig. 2. 5, where an external electric field induces elastic displacemcnt of charges, producing deformation in thc material.
r----------- -----------.
i+- -+--+--;-+--;-+--+--.:;:-+--;-+ ---:
tPolarization
: I
++++++++++
E
t Polarization
1 I
++++++++++++
Direct piezoelectric effect of a piezoelectric element responding to external force
Fig. 2. 4
Converse piezoelectric effect of a piezoelectric element responding to external electric field
Fig. 2. 5
2. Piezoelectric equations The functional intcrrelations bctween elcctric paramctcrs (E, D) and mechanical parameters (T, S) of the piezoelectric effect are known as the equations of state for piezoelectric materials. The forms of the equations are dependent on boundary conditions and indepcndcnt variablcs. The electrical and mechanical boundary conditions of piezoelectric elements depend on the tcsting methods or thcir application circumstances. Strcss-frcc mechanical boundary condition and elamped (fixed) mechanical boundary condition, along with short-circuit and open-circuit electrical boundary conditions, are commonly used in piczoelectric equations. Free dielectric constant and elamped dielectric constant are the dielectric constants mcasurcd undcr stress-frcc mechanical boundary condition and elamped mechanical boundary conditions, symbolized as E~n and E~n' rcspectively. Thc electrical boundary conditions are defined by electrodes, circuit states and geometrical shapes of the piezoelectric elements. During measurement, if the electric field strength E inside the material is maintained to zero but the electric displacement is variable (for example, the electrodes arc shorted or the potential of the sample surface keeps constant by grounding), this boundary condition is called the short-circuit boundary condition. The measured elastic compliance and stiffness arc the short-circuit compliance and short-circuit stiffness, expressed as s~ and c~, respectively. Similarly, if the electric displacement D remains constant but the electric field strength inside the piezoelectric element is variable, the corresponding boundary condition is the open-circuit boundary condition, and the measured elastic compliance and stiffness arc the open-circuit compliance and open-circuit stiffness, expressed as s~ and c~, respectively.
Four typical boundary conditions can be obtained through combining the two mechanical boundary conditions and two electrical boundary conditions, thus leading to four types of piezoelectric equations, which are listed on Table 2. 2.
Ultrasonic Motors Technologies and Ap plicalions
30 Table 2.2 Name
Type 1
Type 2
Type 3
Type 1
Four types of boundary conditions of piezoelectric vibrators
Boundary conditions and corresponding coefficients
Piezoelectric equations with
Loading
Short-circuit compliance coefficients
sE
Short-circuit stiffness coefficients c E
Open-circuit compliance coefficients ,IJ
cD
Clamped impermittivity components
T
j
= =
Si = En
+ E;~nEn
dmJ T J C~Si
enjEn
-
+ E;mEn s;;TJ + gmiDm gnj T + /tnDm
=-
j
T j = Cf,Si - hmjDm
S. D
{f
•
Dm = emiSi
T,D
Stress-free impermittivity components (3T Open-circuit stiffness coefficients
Dm
S,E
ES
Sll bseripts
Si = s5TJ +dniEn
T.E
Stress-free dielectric constants e T
Clamped dielectric constants
shortened
En
=-
hniS i
+ {f",nDm
* i,j=1,2,3,1,5,6; m,n=1,2,3.
The specific forms of the piezoelectric equations listed in the table are also related with the forms of piezoelectric coefficient, elastic compliance and permittivity. As mentioned before, materials with different crystal structures have diffcrcnt indcpcndcnt componcnts in thcir matriccs, so that thc cxprcssions of piczoelectric equations change correspondingly. Furthermore, the piezoelectric cquations may vary cvcn in thc samc piczoelcctric crystal if it is cut into diffcrcnt forms since the piezoelectric coefficient, elastic compliance and dielectric constant will changc in various coordinatc systcms with diffcrcnt rotation symmctrics. For spccific vibration typcs, piczoelcctric cquations may bc furthcr simplificd.
3. Applications of piezoelectric equations zn piezoelectric elements[ll: Here we consider a piezoelectric element as an example of using the piezoelectric cquations. From thc piczoelcctric cquations, it can bc elcarly sccn that thc coupling between electric parameters (E, D) and mechanical parameters (T, S) will inducc elcctromcchanical coupling cffcct in piczoelcctric elcmcnts. Dcpcnding on boundary conditions, thcrc arc two basic piczoelcctric cqua tions in piczoelcctric ccramics. Thc first onc is thc piczoelcctric cquation dcscribing thc dircct piczoclectric effect under short-circuit electrical boundary conditions (E 1 = E2 = E, = 0). If transforming thc piczoelcctric cquation Dm = d mj T J +E~nEn (listcd in Tablc 2.2) into matrix form, we have
D=
rZl~~ r D3
d" d'l d'l
d l, d 22 d"
d 13
dl1
d 15
d'3 d"
d" d34
d'5 d,s
T1 T,
d"1 d'6 d'5
T3 Tl T,
E,
E3
0)
T6 (2. 12) In the equation, D" Ei(i=l, 2, 3) are the electric displacement and electric field in thc polar planc of thc piczoelcctric body along Xi (i = 1, 2, 3) axcs. Thc
Chapter 2
Fundamentals of Piezoelectricity and Piezoelectric···
31
electric displacement vector D = [D l D2 D 3 ]I'; d is a matrix retrieved from the reduced piezoelectric constant tensors. where the component of piezoelectric constant d i} signifies the electric displacement developed along the i-axis when the material is stressed along the j-axis. The form of matrix d is determined according to the specific material and its crystal structure. The second one is the piezoelectric equation describing the converse piezoelectric effect at stress-free mechanical boundary conditions (Ti = O. i = 1-6). By the piezoelectric equation 5 i = s~T} + dmEn listed in Table 2. 2. the electric field E3 along the .1':3 axis. and electric fields El and E2 along axes of .1':1 and .1':2 will generate strains in piezoelectric bodies. This is the converse piezoelectric effect.
s=
51 52 53 51 55 56
=
dTE
=
dJJ d 12
d 21
d 31
d 22
d 32
d 13
d23
d 33
du
d"
d 15
d" d 26
d 16
d 35
[E' E2
(Ti
= O.i =
1.2.··· .6)
E3
d 36
(2. 13) In piezoelectric elements. the polarization direction is usually referred as .1':3 axis. For piezoelectric ceramics with polaxis parallel to X3 • the plane perpendicular to the polaxis is isotropic. so that there are only three independent piezoelectric constant components d 31 • d 33 • and d 15 in piezoelectric constant matrix. The piezoelectric constant matrix can be written as
d=
[1,
0 0
0 0
0
d 13
d 13
0
d 31
d 33
0
0
~1
(2. 14)
Meanwhile. at the stress-free boundary conditions. piezoelectric ceramics can be regarded as capacitors so that they should satisfy the dielectric equations. Similarly. under the condition of short circuit. piezoelectric ceramics also need to comply with the constitutive equations. These equations are not listed here.
2. 4 2. 4. 1
Vibration Types of Piezoelectric Vibrators Piezoelectric Vibrators and Their Equivalent Circuits
Piezoelectric vibrator is simply a piezoelectric bulk with electrodes coated on its two opposite surfaces. which is the most elementary piezoelectric unit used in ultrasonic motors or other actuators. As an elastic body. it has infinite natural vibration frequencies. Once the frequency of the applied electric field equals to one of the natural frequencies. mechanical resonance will be activated in the vibrator due to the converse piezoelectric effect. Vibration types are defined by the relations between polarization directions and vibration directions. We can refer to the type as the longitudinal vibration type if the vibration direction is parallel to the polarization direction. For a transverse vibration type. the vibration direction
Ultrasonic Motors Technologies and Ap plicalions
32
is perpendicular to the polarization direction. Table 2. 3 lists several vibration types observed from some regular vibrators, where the hollow arrow represents the polarization direction and the solid arrows point to the vibrating directions. Among them, diagrams (a) to (g) show length-extension(contraction) types (The extensions(contractions) may be along length, width or radial directions) whose vibration is perpendicular to the polarization direction (LE type). Table 2. 3 (h) presents a transverse-thickness shear vibration type(TS type) in a piezoelectric plate whose vibration is normal to the polarization direction. As a comparison. a longitudinal-thickness shear vibration type is given in Table 2.3 (1) (LS) , where the vibration is parallel to the polarization direction. Those thickness-extension ( contraction) vibrations parallel to polarization directions arc shown in Table 2. 3(i)-(k) (TE types). Table 2. 3
---( ~.,
I~
(a) Length exten ion
a
Vibration types of piezoelectric vibrators
(b) Width exten ion
~
'c"'
1:;
r:
I-
~O ~O (e) Radial exten ion
U
~ .,
<--/
""iii
~
" 'a, c 0
..J
0
V (i) Thickness exten ion
(I) Radinl extension
0
(c) Width exten ion
u
(g) Radial exten ion
-(
0
~
(d) Length exten ion
..~
L
0
¢o ~ 0 0
(j) Thickness extension
(k) Thickness extension
~
(h) Thickness shcnr
0
(I) Thickness hear
Equivalent circuits are usually employed to characterize the electric properties of piezoelectric vibrators. For example. the piezoelectric bar vibrator shown in Table 2. 3(a). in which an one-dimensionallength-extension(eontraetion) vibration is produced due to the transverse effect. its equivalent circuit can be described by Fig. 2. 6 (a) from the electric-mechanical simulation analysis. In the figure, CD denotes the elamped capacitance; C i , L i and Ri represent the equivalent capacitance, inductance and resistance of the ith order modal stiffness, mass and damping. where C, equals C1 / i 2 (i is odd). It should be mentioned that the loss of the piezoelectric vibrator is also taken into account in the equivalent circuit. On the other hand. for the longitudinal vibration type. e. g.• the piezoelectric disk vibrator shown in Table 2. 3 (k), where the vibration along the thickness direction is caused by the longitudinal effect, the equivalent circuit can be depicted by Fig. 2. 6(b).
Chapter 2
1st
Fundamentals of Piezoelectricity and Piezoelectric···
3rd
5th
i
th
(a) Equivalent circuit of the piezoe lectric vibrator working with transversal eflect
Fig. 2. 6
2. 4. 2
33
(b) Equivalent circuit of the piezoelectric vibrator working with longitudinal effect
Equivalent circuits of piezoelectric vibrators
Characteristic Frequencies of Piezoelectric Vibrators llO -
If voltage signal with adjustable frequency is applied to the equivalent circuit eIther in Fig. 2. 6 (a) or (b), the dependence of the impedance of the vibrator on the signal frequency can be schematically described by the curve in Fig. 2. 7. The frequency that the impedance reaches its minimum is the minimal impedance frequency, symbolized by Iml . With the increase of input signal frequency the vibrator then reaches the maximal impedance frequency I nl. In electrics Iml and Inl are defined as the first order resonance frequency and anti-resonance frequency. respectively. When further increase the input signal frequency, a series of maxima and minima will appear periodically, which correspond to the ith order resonance frequency Imi and anti-resonance frequency Ini (i = 1 ,2,3···).
"
-g" ::
Q.
.§
~
IZI~
--------
oL-------------~~--~~~--~~----~f
Fig. 2. 7
Electrical impedance of piezoelectric vibrator as a function of frequency
Considering the corresponding equivalent circuit, for example, )Jear the first order resonance frequency. the equivalent circuit of piezoelectric vibrators with losses can be simplified by Fig. 2. 8, where Ll is the equivalent inductance (also called the dynamical inductance of the resonance). which is correlated with the material mechanical properties and is analogous to the vibrator mass; Rl is the
Ultrasonic Motors Technologies and Ap plicalions
34
equivalent resistance related to the mechanical loss; and C j is the dynamic capacitance.
n Cl)C'
~R'
the static capacitance
H
G
x,
R,
Fig. 2. 8
CO IS
Equivalent circuits of piezoelectric vibrator
The admittance of the equivalent circuit can be described as
Y = iwC o +
.
Rl
1
1
+ IwLl + :---C Iw
= Yo
+Y
d
= G + iB
(2. 15a)
j
where Yo and Yd are the static and dynamic admittances, respectively. G and B are the real and imaginary parts of the admittance. The impedance of the equivalent circuit can be described as Z=~=_i_.
Y
=j:
wC o
n-iO' 1- n+ iO' = R
. + IX,
(2.1Sb)
where R, and X, are the real and imaginary parts of the impedance, respectively.
n= ~
I>
is a normalized frequency factor; 0'
=
21(ICo R is a normalized damp-
,
ing factor. I, and Ip are series and parallel resonance frequencies, respectively. We can still obtain the general relationship between the electrical and mechanical domains by means of the electric-circuit analogy:
F
=
Zmv- X:V
(2. 16a)
+ YoV
(2.16b)
1= x:v
where V and I arc the voltage and current at the electrical port, respectively. F and v arc the force and velocity at the mechanical port, respectively, and x: is the force coefficient of the piezoelectric component, Zm is the mechanical impedance of the vibrator. We take the conductance (the real part of the admittance) as the horizontal ordinate and the susceptance (the imaginary part of the admittance) as the vertical ordinate. When the frequency varies in the vicinity of the first order resonance frequency, the moving trace of Y d will form a circle, with the center at (l/2R j , 0) and radius of 1/2R 1 • As Y d phase vector moves a whole circle, the phase vector of Yo 01 Ro) has little change so that it is treated as a constant. Consequently, the moving track of Y phase vector, which forms the so called admittance circle, can be obtained by shifting Y d circle along the vertical and horizontal coordinates with w,Co and II Ro in the complex plane, as shown in Fig. 2. 9. Some important parameters in the equivalent circuit of piezoelectric vibrators
Chapter 2
Fundamentals of Piezoelectricity and Piezoelectric···
35
can be calculated usmg the admittance circle. From the diagram three pans of resonance frequencies can be observed. which arc summarized in Table 2. 4 for the convenience of comparison. B
\fRo
G
IIRr Fig. 2. 9 Table 2. 4 Characteristic
Diagram of admittance circle
Characteristic frequencies of piezoelectric vibrators Explanation
Character
fm
Minimum impedance frequency
~=o
fn
Maximum impedance frequency
~=o
f,
Series resonance frcq ueney
Xl = 0
[]=o
fp
Parallel resonance frequency
Xc = CD RJ =0
[]=l
f,
Resonance frequency
f,
Anti-resonance frequency
Frequency
dw
dw
Frequency equation ([]2 +(2)' _
282 ([] + y)
- 2[]yO - []) - []2 = 0 ([]2 +(2)2 -
282 ([] + y)
- 2[]y(l - []) - []' = 0
Xc = 0 Bp = 0 Xc = 0 Bp = 0
[]O-[]) -8 2 = 0 [](l-[]) -8' = 0
These characteristic frequencies are indispensable to determining the properties of a piezoelectric vibrator. It has been verified that in most cases we have the following equations by reasonable approximation
2. 4. 3
fm
=
f,
=
f,
(2. 17)
fn
=
f,
=
fp
(2. 18)
Coupling Coefficient and Quality Factor
1. Electromechanical coupling coefficient The electromechanical coupling coefficient is a key parameter that describes the coupling degree between mechanical energy and electrical energy in piezoelectric materials. The definition is written as
36
Ultrasonic Motors Technologies and Ap plicalions
k'
=
Mechanic energy created by converse piezoelectricity Input electric energy
(2. 19)
k'
=
Electric energy created by direct piezoelectricity Input mechanic energy
(2. 20)
or
Let's think a piezoelectric element exerted with a stress or electric field, if we usc WI and W, to represent the elastic energy density and the dielectric energy density of a unit volume, and WI' to represent the elastic-dielectric interaction energy of the unit volume(also called as piezoelectric energy). the electromechanical coupling coefficient of this unit volume then can be defined as (2.21) In other words. the electromechanical coupling coefficient equals the ratio between the elastic-dielectric interaction energy and the geometric mean of the dielectric energy density and the elastic energy density. The mechanical energy of a piezoelectric vibrator depends on its shape as well as the vibrating type. Therefore. for different vibrating types. particular electromechanical coupling coefficients have to be used. For example. k" is referred to as the planarCin-plane) electromechanical coupling coefficient for the radial vibration type in a piezoelectric disk. As we mentioned. the polarization direction of piezoelectric elements is always referred to as z axis. so that k'l gives the transverse electromechanical coupling coefficient. signifying the length-extension vibration type of a piezoelectric plate; k" , referred to as the longitudinal electromechanical coupling coefficient. expresses the thickness-extension vibration type of a piezoelectric bar; k , • which characterizes the thickness-extension vibration type of a thin piezoelectric plate, is defined as the thickness-extension vibration type electromechanical coupling coefficient; and k l5 is the thickness-shear vibration type electromechanical coupling coefficient. depicting the thickness-shear vibration type of a rectangular plate. These five vibrating types and their corresponding electromechanical coupling coefficients are listed in Table 2. 5.
2. Mechanical quality f actor[l, 10: Mechanical quality factor Qm denotes the energy consummg of the piezoelectric element to overcome the inner friction during its resonance. It is an important parameter to characterize the performance of piezoelectric materials. referred to as the quotient between the mechanical energy stored and the mechanical energy lost within a period of a resonant vibration. Devices such as filters. resonance type transducers and standard-frequency vibrators primarily base on the mechanical resonance effect of piezoelectric vibrators. Qm is then defined as:
Q m
=
2 IT
Mechanical energy stored during a period in resonance (2 22) Loss of mechanical energy during a period in resonance .
Chapter 2
Table 2.5
37
Fundamentals of Piezoelectricity and Piezoelectric···
Five electromechanical coupling coefficients of piezoelectric vibrators Electromechanical coupling coefficients
Sample shapes/vibration types
Planar electromechanical coupling coefficient kp
k' = _2_ d;l
1-
p
(J
Srl EI~
Transverse electromechanical coupling coefficient -<:
k31
d~l
2
k31=~
~~------------~
S11 £33
PC:> Longitudinal electromechanical CDupling
Electrode plane
coefficient
kL
=
k33
d2 l~ 3~I 533£33
Thickness extension type
electromechanical coupling coefficient k t
,
k 2t
Displacement direction
e33
cr~ E~3
Thickness shear type electromechanical coupling coefficient k 1S
PC:;> Obviously, the parameter Qm of a vibrator is also dependent on its vibration type. Here Qm is the mechanical quality factor of the radial vibration type if without special clarification. From the equivalent circuit, we can determine Qrn by the cleetrieal parameters of the vibrator
1
Qrn
=
4rr(Co
+ C] )R] t::.j
(2. 23)
Ultrasonic Motors Technologies and Ap plicalions
38
Co: the static capacitor of the piezoelectric vibrator; R I : the equivalent resistance of the vibrator in the resonant state; C I : the equivalent capacitor of the vibrator in the resonant state; 6.j: the difference between the resonant frequency j, and the anti-resonant frequency j,.
2.5
Applications of Piezoelectric Materials to Ultrasonic Motors
Piezoelectric materials playa key role in ultrasonic motors and other piezoelectric actuators because of their function to transform electrical energy into mechanical energy. Several parameters of some important piezoelectric materials are listed in Table 2. 6 for reference- IIJ • Since the properties of piezoelectrics can be widely adjusted by substituting or doping additives, the data show only a rough range. In the table, To is the Curie temperature and Eo is the coercive field. Table 2. 6
Some main parameters of several typical piezoelectric materials Parameters
Materials
T,
E§3
rC
d 33
diS
I (pC/")
l(pC/")
k33
Qm
E, l(kV lern)
Pb(Zr, Ti)03 (PZT)
330
1 800
117
710
0.73
75
(BaPb) Nb z 0 3 (BPN)
100
300
85
100
0.30
15
PbTi 0 3 (PT)
490
190
56
68
0.45
1 300
Bi4 Ti4 Ole (NBT)
600
110
18
-
O. 15
100
(Bio. 5 Nao. s) Ti0 3 (BNT)
315
300
70
-
O. 10
210
73
LiNb0 3 (L") Crystal
1 150
25
6
69
0.23
NR
200
SiO, (Quartz) Crystal
573
4.5
2(d ll )
NR
10 5
-
10-12 -
> >
40 50
-
The requirements in properties of piezoelectric materials have to be determined according to their specific purposes of the devices. These used in ultra-high-frequency (UHF) and high-frequency devices require the material to have low permittivity and small high-frequency dielectric loss. For energy transducer application, the coupling coefficient and acoustic impedance of the material arc often stressed. Materials with excellent frequency stability and high Qm values can be used as standard frequency oscillators. To satisfy the application of delay line, the materials have to be stable in frequency, and the velocity of sound in the materials should also be considered. Ceramics used in the electro-acoustic field should have a large permittivity, high kp value and high elastic compliance coefficient, and their dielectric loss doesn't matter too much to the devices. For hydroacoustic transducer applications, if used as receivers, it is necessary that the material has a large piezoelectric coefficient of g33 or g31 , large permittivity, high kp value and high compliance constant, but its Qrn value is not seriously required; if
Chapter 2
Fundamentals of Piezoelectricity and Piezoelectric'"
39
used as high-power emitters, it is important that the material has a low dielectric loss tanO' and a high Qrn undcr a rong elcctric field, additionally with a largc diclectric constant, high kp value and large piezoelectric constant. For materials uscd as filtcrs, thcy arc cxpcctcd to bc not only with cxccllcnt duration stability and temperature stability, but also with high Qm and low tanO'; the requirement in kp valuc dcpcnds on thc bandwidth of filtcrs. High-voltagc gcncrators and igniters require the materials to have large values of g33 and k33' a large permittivity, a moderately high Qm' as well as a low tan 0'. :'\Jowadays, by means of doping and substituting, thc propcrtics of piczoelcctric ccramics can bc adjustcd in a wide range to meet diverse application occasions.
2.5. 1
Piezoelectric Ceramics Used for Ultrasonic Motors
1. Piezoelectric ceramics for ultrasonic motors So far, piezoelectric ceramics, instead of piezoelectric single crystals, are mainly thc functional matcrials uscd in ultrasonic motors. Among thcm, Pb(Zrx TilE) 0, (PZT) bascd systcm is thc most important onc duc to its cxtraordinary propcrtics and is currcntly thc first choicc for ultrasonic motors. The preparation of PZT ceramics follows a standard ceramic process ineluding stcps of powdcr prcparing, forming and sintcring, succccdcd by a poling proccss which is requisite for piezoelectric ceramics to get piezoelectricity. Ulltrasonic motors are typical high-power devices, so that the used piezoelectric ccramics should bc with a low dielcctric loss tani) and a high mcchanical quality factor Qm under a strong electric field, as well as a reasonable piezoelectric constant d 33 and an elcctromcchanical coupling factor k p • Unfortunately, for PZT-based piezoelectric ceramics, in most cases the improvement in Qrn simultaneously induces degradations in d" and k p • Up to now, researchers from all over thc world havc conductcd a grcat dcal of fruitful work, which aim cd at cnhancing the piezoelectric constant and electromechanical coupling coefficient but without impairing the mechanical quality factor. Multi-constituent doped PZT ceramics havc bccn dcvelopcd for high powcr piczoelcctric dcviccs, and thc improvcmcnt in properties still continues. Ultrasonic motors now have been used in tremendous occasions to satisfy many applications. For different motors with specific purposes, piezoelectric elements wi th particular properties, dimensions and structures are required. The piezoelcctric elcmcnts thcn havc to bc shapcd into spccific dimcnsions and forms. Fig. 2. 10 (a) shows somc PZT piczoelcctric elcmcnts from industrial companics, and Fig. 2. 10 (b) displays PZT components used for ultrasonic motors designed by our PDLab at :'\JUAA. 2. Stability of Piezoelectric ceramics used in ultrasonic motors[1-2, 11-l2J During thc scrvicc of ultrasonic motors, thc uscd piczoelcctric matcrials havc bccn found fluctuant in thcir physical propcrtics with elapsc of durations and fluctuation of tcmpcraturcs. Somctimcs thc variation in propcrtics can bc pro-
40
Ultrasonic Motors Technologies and Ap plicalions
(a) PiezoeleCTric ceramic elemCIlIs(Xinchang Silver River Electronic Co. Lid in China)
Fig. 2. 10
(b) PZT components lISed in ultrasonic molors developed by PDLab
PZT piezoelectric ceramic components
nounced enough to cause failure of the whole devices. Therefore, the stability of piczoelectric ccramics is utterly significant for thcir applications. (1) Aging stability Stabilities of piezoelectric ceramics with duration elapse and temperature fluctuation are called aging stability and tcmperaturc stability, rcspectively. Variations of physical propertics induccd by aging effcct will accumulatc in thc polarizcd ccramics at a gradually slowing-down ratc. Thc accumulation is irrevcrsible unlcss thc ccramics cxperience a ncw cxcitation such as a rcpolarization. Generally, as a result of aging, samples present decreases in dielectric constant, dielectric loss, piezoelcctric coefficicnt and elastic compliance and increases in mechanical quality factor and frequcncy factor. It is also found that such changes are roughly proportional to the logarithm of the duration. The aging curvc of the pol cd piezoelcctric ccramics may bc interfercd by environmental factors. To deal with possible disturbances, a prior artificial aging treatment is usually employed to stabilize the poled ceramics so that the ceramics won't fluctuate in properties with the environmental interferences. In practical manufactures, the poled piezoelectric ceramics are aged by heat treatment or heat cyeles. The prior heat treatment can help the ceramics to stabilize from other heat excitements. This stabilization is attributed to the mechanism that domain motions of the ceramic are enhanced and its internal stress is largely released during the artificial aging processing. Other artificial aging methods such as elcctric aging, mechanical aging and exposure to y-ray radiation of Co", can also achicvc similar cffects. (2) Thermal depolarization Thermal depolarization will occur during heating the piezoelectric ceramIcs. Dipolcs get in disorder gradually with the tempcraturc elevating, dcteriorating the piezoelectric performance simultaneously. Once the temperature reaches abovc the Curic point, whcre thc piczoelectricity disappcars thoroughly, the ultrasonic motor is irreversiblc dcstroyed. Consequcntly, it is ncccssary that thc divices operatc at tcmpcraturcs far below the Curie point. Thc tcmpcraturc limitation whcrc piezoelectric ccramics can safely work without rcmarkablc reduction in piczoelectric activity is approximately set at half of thc Curie point.
Chapter 2
Fundamentals of Piezoelectricity and Piezoelectric···
41
(3) Electric depolarization Electric dcpolarization happcns if a strong invcrse electric field is applicd on a polcd ccramic. Whcther the elcctric field will cause scrious dcpolarization in thc material depends on the material itself and the applying duration of the electric field. as well as the temperature that the material stays. For a direct current electric field. thc strcngth thrcshold causing dcpolarization is around 5001 OOOv Imm. Thc readers have also to notice that thc material can bc depolarizcd during the other half period when applying an altering current field to drive ultrasonic motors. (4) Mcchanical dcpolarization Excessive mechanical strcss can disarrange dipolcs in piczoelectric ceramIcs. leading to a failure in piczoelectric pcrformancc. This process is referrcd to as mechanical depolarization. Piczoelectric ceramics with differcnt compositions may allow different limitations of the safe mechanical stress. Reliable data and information have to be referred for reasonable application of materials.
2. 5. 2
Applications of Piezoelectric Materials to Other Actuators
Besides ultrasonic motors. other piezoelectric actuators based on different forms of piezoelectric materials havc also been widely applied in many fields. Considcring their application importancc and their distinct opcrating mcchanisms in contrast with ultrasonic motors, we will then launch a brief introduction for these piezoelectric actuators.
1. Piezoelectric stack actuators In conventional piezoelectric actuators. the displacements of single layer piezoelcctric actuators are found to bc too small to fulfilllargc strokc driving. Thc idca thcn comes naturally to stack several piezoelcctric ceramic pieccs together to form a piezoelectric stack actuator. As shown in Fig. 2. 11, the piezoelectric stack actuators are fabricated by agglutinating piezoelectric ceramic pieces in serics. Thcse picces are electrically parallel but mcchanically scrial. When a voltage IS applicd along the poling direction. each singlc picce produces a displacement, and all displacements sum up to the total output of the stack actuator.
Fig. 2. 11
Piezoelectric stack actuators [PI (Physik Instrument) L. P. ]
Current concerns for piczoelectric stack actuators are mainly on to compensatc thc hysteresis characteristic of the dcviccs. This bchavior refcrring to thc nonlin-
42
Ultrasonic Motors Technologies and Ap plicalions
ear hysteresis between the input voltages and the output displacements, is an intrinsic trait of piezoelectric materials. Therefore, some compensation methods have to be adopted to improve the positioning precision of the actuators.
2. Piezoelectric bimorph actuators Piezoelectric bimorph was firstly invented by Baldwin Sawyer in 1931. Now it has been frequently used in piezoelectric elements for acoustic detections, USMs, laser beam deflectors, filters, accelerometers, optical choppers, etc l3J • There are four structures commonly used in piezoelectric bimorphs, whose schematic diagrams arc presented in Fig. 2.12. In diagrams (a) and (b) two identical piezoelectric plates arc bonded to each other, with their poling directions oppositely arranged. Electrodes are coated on both sides of the bimorph. These two structures arc therefore called antiparallel-type piezoelectric bimorphs or continuous-type piezoelectric bimorphs. The bimorph in Fig. 2. 12 (c) contains an extra electrode between two plates, and both plates are poled along the direction of the driving voltage. Contrastively, this structure is named as parallel-type piezoelectric bimorph. The actuator in Fig. 2. 12 (d) consists of a non-piezoelectric plate and a piezoelectric vibrator coated with electrodes.
{jJ U ~
(a)
f~
(e)
Fig. 2. 12
{jJ f[
t~ (b)
ff' (d)
Structures of piezoelectric bimorphs
When applied with an electric field, owing to the converse piezoelectric effect, the bimorphs in (a), (b) and (c) start to bend since one vibrator inside the structures extends and the other contracts. The working voltage of the bimorph in (c) is twice as large as those of the bimorphs in (a) and (b), so the bending deforma tion of the bimorph in (c) is also twice as large as those of the bimorphs in (a) and (b). The behavior of the fourth one in (d) is similar with that of the former three ones, but its motion and output is tunable by varying thickness ratios between the piezoelectric plate and the non-piezoelectric one or by changing the elasticity modulus of the non-piezoelectric material.
3. Functionally graded piezoelectric actuators For functionally graded piezoelectric materials, the composItions and structures are controlled to change gradually from one side to the other side. Correspondingly, the properties and functions of the materials change gradually too. A structural comparison between the bimorph and the functionally graded device is shown in Fig. 2. 13. In bimorph structure, the sharp interface is easy to concen-
Chapter 2
Fundamentals of Piezoelectricity and Piezoelectric···
43
trate stress and induce interface cracks. However, these disadvantages can be effectively avoided in the gradient interface of the functionally graded material. + v --~
________________________~
-v --~
________________________~ (a) Piezoelectric bimorph
(b) Functionally graded piezoelectric material
Fig. 2. 13 A comparison between the structures of piezoelectric bimorph and functionally graded piezoelectric material
Figure 2. 14 shows the working mechanisms of a multi-layer functionally graded piezoelectric actuator[14:. Four layers with different compositions and properties are sintered together to form a structure without evident interfaces. The piezoelectric coefficients and dielectric constants vary from bottom to top in a positive and negative gradient. respectively. Under a DC voltage in thickness direction. the layers of smaller dielectric constants will have larger electric-field intensity distributions. As a result, each layer deforms in the way as the Fig. 2. l1(b) shows. The deformations then integrate into a uniform deformation in the whole device. as shown in Fig. 2. 14(c). Apparently, the internal stresses are greatly depressed in the device owing to the structural gradient. Furthermore, higher mechanical strength can be obtained for the absence of adhesive between the interfaces.
(a) Schematic structllre ofa functionally graded piezoelectric actuator with four piezoelectric layers
(b) The defomlation of each layer under applied voltage
(c) Total defomlation of the fu nctionally graded piezoelectric aCllmtor
Fig. 2. 14
Deformation mechanism of the functionally graded piezoelectric actuator
4. Piezoelectric fiber actuators The brittleness of piezoelectric ceramics restrains their applications in nonplanar devices. For this reason, active fiber composites (AFe) and macro fiber compos-
Ultrasonic Motors Technologies and Ap plicalions
44
ites (MFC) were designed by American scientists from MIT and NASA in the 1990' s. These composites arc generally called piezoelectric fiber compositesL 15J . In AFC structures, arrays of piezoelectric fibers with round cross sections arc embedded in epoxy resin matrix. Interdigital electrodes are arranged perpendicularly to the axial direction of the fibers. The fabrication of MFC is analogical to that of AFC except that the piezoelectric fibers inside arc with rectangular cross sections. Large stains can be obtained in AFC and MFC by utilizing their axial d" piezoelectric characteristics. Comparing with piezoelectric ceramics, piezoelectric fiber composites possess better flexibility so that they can satisfy the applications in bending planes(Fig. 2.15).
Piezoelectric active fiber composites (left) and macro fiber composites (right)
Fig. 2. 15
Recently. metal core piezoelectric fibers (MPF) are developed to fabricate new piezoelectric sensors and actuatorS[16 17J. A typical structure of MPF is shown in Fig. 2. 16, where the piezoelectric fiber with diameter O. 15-0. 25mm is coated with a layer of metal electrode on the surface. In the center of the piezoelectric fiber, a metal core of O. 05mm in diameter, usually platinum. acts as another electrode. This metal core can also act as a medium that enhances the strength of the fiber. Metal COre Piezoelectric cCl'1lmic
T/
..... -
I
I
I
1 - .....
J-_I_
~-J
'1
~
Fig. 2. 16
Piezoelectric ceramic fiber with a metal core inside
We refer the fiber to as the full-electrode piezoelectric fiber if its surface is entirely coated with metal electrode. The piezoelectric fiber is polarized along the radial direction, so that the fiber will produce a radial extension vibration type under an applied electric field. A half-plated fiber is called the half-electrode piezoelectric fiber. Bending distortions will be produced in the half-electrode piezoelectric fiber after applying electric field. These tiny dimensional MPF can be con-
Chapter 2
Fundamentals of Piezoelectricity and Piezoelectric···
45
veniently embedded into composites to drive deformations in the composites.
2.6
Advances in Novel Piezoelectric Materials
With the fast-growing application of ultrasonic motors and other piezoelectric devices, the demand for new piezoelectric materials is getting more and more imperative. Here we present some progress in the research on piezoelectric materials that are promising for future potential applications.
1. Multi-constituent doped PbCZr x Ti 1- x )()3CPZT) ceramics[1820: Multi-constituent doped PZT ceramics have been intensively investigated for long duration since its electrical and mechanical properties of the ceramics can be greatly varied by doping PZT with acceptors, donors, or isovalent dopant. It was verified that doping of Mn' I in PZT could improve the mechanical quality factor Qm; and its electromechanical coupling coefficient k" could be improved by Sb 5 - doping. So that PZT based ceramics doped with both Mn'- and Sb'- have been constantly attended. By these efforts some trinary and quaternary PZTbased materials with excellent sintcring property and repeatability have been found. Materials applied for high-power piezoelectric devices have been reported based on these systems. For example, in Mn'· doped quaternary PS:'\J-PZ:'\J-PZT system, experiments reveal that pure perovskitc phase can be formed within a wide range of Mn' I additives. Proper amounts of Mn'· additives can optimize the piezoelectric properties of the PS:'\J-PZ:'\J-PZT quaternary system for improving the mechanical quality factor Qm and decreasing the dielectric loss tanB. The ceramics with o. 5 % (mass fraction) dopant possess the best electromechanical properties that satisfy the requirement for USM and transformer applications. Recently a new series of quaternary PZT-based ceramics doped with Ba" and Sr'· have been developed with properties greatly improved. Among them, d 33 = 406pC/:'\J, kp
=
o. 55,
E=
2183, Qm = 1077, and tani) = 2. 7 %, respectively.
2. Relaxor ferroelectric single crystals[5] Relaxor ferroelcctrics such as lead magnesium niobate-lcad titanate (PM:'\J-PT) and lead zinc niobate-lead titanate (PZ:'\J-PT) single crystals exhibit much higher piezoelectric coefficients and electromechanical coupling coefficients than conventional piezoelectric ceramics- 5-. For example, the strain of these crystals reaches to 1. 7 %, almost an order larger in magnitude of the conventional piezoelectric ceramics. Furthermore, it has been verified that even temperature down to -200'C, the property of PMN-PT and PZN-PT single crystals are still comparable with the room-temperature property of PZT ceramics['l 24:. This nature enables PMN-PT and PZN-PT suitable for USM running at extreme-temperature conditions. Relaxor ferroelectric single crystals are promising to replace conventional piezoelectric ceramics in many devices, such as acoustic detectors, ultrasonic imaging devices, high-strain actuator, and ultrasonic motors used in extreme circumstances- 25 - 31J •
3. Lead- free piezoelectric materials[35] The lead-containing piezoelectric ceramics may cause serious hazard to the envi-
46
Ultrasonic Motors Technologies and Ap plicalions
ronment and human health during their manufacturing, serving and disposing after failure. Therefore the development of environment-friendly lead-free piezoelectric ceramics is indispensable from the perspective of the global environment protection. Recent research on lead-free piezoelectric materials has been focused mainly on two promising systems: perovskite structural piezoelectric ceramics and bismuthlayered structural piezoelectric ceramics. The former family ineludes the solid solutions of Ko 5 :'\lao. 5 :'\Ib03 - LiTa0 3 , BaTi03 - Bio 5 Ko. 5 Ti0 3 , and Bio. 5 Nao. 5 Ti0 3 Bio. 5 Ko. 5 TiO, , with their compositions near the morpho tropic phase boundary. In optimized compositions the piezoelectric constant d 33 of these systems can reach 300pC/:'\I, which is elose to the value of PZT ceramics. The bismuth-layered structural solid solutions such as the donor-doped Bi4 Ti3 0 12 or Bi, TiTaO g systems are featured with high Curie point and relatively large piezoelectric coefficient, as well as the less temperature dependence of their resonant frequencies. These traits make them suitable for sensor and resonator applications.
4. Piezoelectric composites[6, 18: Piezoelectric composites have been developed since the late 70s of last century. The preparation of piezoelectric composites involves to incorporate piezoelectric ceramics and piezoelectric polymers with designed connectivity, mass/volume ratios and spatial distributions to form certain microstructures. Piezoelectric ceramics have disadvantages such as high density, extreme brittleness, easy fracture to mechanical impacts and poor capability to form complex shapes. On the other hand, piezoelectric polymers possess properties of excellent flexibility, low density and great machinability but poor temperature endurance and high cost. However, the properties of piezoelectric composites can be remarkable improved by taking advantage of the composition effect elaborately, so that piezoelectric composites keep the merits of both components of piezoelectric ceramic and polymer and overcome their disadvantages, offering excellent piezoelectric performance and mechanical flexibility. The manner of each phase connects with itself in composites is known as the "connectivity" of the composites, which is proposed by Newnham et al. in 1978. Fig. 2. 17 lists all ten types of connectivity of piezoelectric composites. These connectivity types are: 0-0, 0-1, 0-2, 0-3, 1-1, 1-2, 1-3, 2-2, 2-3, and 3-3. The first number in the expression represents the connecting dimension of the piezoelectric phase and the second number is the connecting dimension of the polymer phase. Different connectivity types mean different spatial distributions of the ceramic phase and the polymer phase and correspondingly different dielectric and piezoelectric properties in the composites.
5. Piezoelectric thin jilms[36 37J The progress in thin film deposition methods has provided the possibility of application thin films in almost all fields of modern science and technology. Now bunch of techniques have been employed to prepare piezoelectric materials from high-quality epitaxial films to large-area polycrystalline films. Among them,
Chapter 2
Fundamentals of Piezoelectricity and Piezoelectric'"
---__ ---0-0
0- I
0-2
0-3
I- I
1-2
2 -2
2 -3
1-3
Fig. 2. 17
47
3-3 (Two views)
Ten connectivity types of piezoelectric composites
sputter deposition, sol-gel, chemical vapor deposition (CVD), molecular-beam epitaxy (MBE) and pulsed laser deposition (PLD) are well established for piezoelectric films preparation. Piezoelectric films could have many important applications due to their versatile properties. For example, in surface acoustic wave ( SAW) devices. piezoelectric films have been widely used as the functional parts. The combination of micro sensors and actuators onto the surfaces of semiconductor integrated circuits creates a new research highlight of piezoelectric film micro mechanic-electric systems. Devices based on bulk piezoelectric materials usually operate with operating frequencies no more than hundreds hertz due to their dimension restrictions. On the other hand. devices based on piezoelectric films offer much higher operating frequency, extra flexibility in designing and shaping the device dimensions, as well as additional advantage in device miniaturization and integration. In various applications, piezoelectric films can replace their single crystal or ceramic counterparts, to provide similar functions with considerable satisfaction.
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Zhiwcn Yin. Physics of Dielectrics (Second Edition). Bcijing: Prcss, 2005:1-8. (in Chincsc) B Jaffe, W R Cook, H Jaffe. Piezoelectric Ceramics. "few York: Academic Press, 1971: 1-5. Duan Fcng, Changxu Shi, Zhiguo Liu. Introduction to Material Science-An Integrated Approach. Beijing: Chemical Industry Press, 2002:324-350. (in Chinese) B Jaffe, R S Roth, S Marzullo. Piezoelectric properties of lead zirconate-Iead titanate solid solution ccramies. J. Appl. Phys., 1951,25: 809-100. R F Service. Shape-changing crystals get shifter. Science, 1997, 275: 1878. Changxu Shi, Hengde Li, Lian Zhou. Handbook of Materials Science and Engineering. Beijing: Chcmieal Industry Prcss, 2006: 7-76. (in Chincse) Shenghe Lin, Zhibi Ye, Yubin Wang. Piezoelectric Ceramics. Beijing: Defense Industry Press, 1980: 17-40. (in Chinese) Statc burcau of technical supcrvision. National Standards of the People's Republic of China CElT 3389. 1-1996. Bcijing: Standards Prcss of China, 1997: 2-3. Daorcn Song, Mingshan Xiao. Piezoelectricity and Its Application. Bcijing: Popular Sciencc
Ultrasonic Motors Technologies and Ap plicalions
48
[IOJ [l1J [I2J [13J [I1J [I5J [16J
[17J [I8J
[I9J
[20J [21J [22J [23J [21J [25J [26J [27J
[28J [29J [30J
Press, 1980:19-38. (in Chinese) Yuan Li, Zikai Qin, Zhigang Zhou. Measurement for Piezoelectric and Ferroelectric Materials. Beijing: Science Press, 2001:19-21. (in Chinese) Yuhuan Xu. Ferroelectric and Piezoelectric Materials. Beijing: Science Press, 1978: 118202. (in Chinese) Weilie Zhong. Physics of Ferroelectrics. Beijing: Science Press, 1996: 310-311. (in Chinese) J G Smits, S I Dalke, T K Cooney. The constituent equations of piezoelectric bimorphs. Sensors and Actuators A, 1991, 28: 41-61. J Qiu, J Tani, Ueno, et al. Fabrication and high durability of functionally graded piezoelectric bending actuators. Journal of Smart Materials and Structures, 2003, 12: 115-12l. R B Williams, G Park, D J Inman. An overview of composite actuators with piezoelectric fibers. Proc. of SP IE- The International Society of Optical Structures, 2002, 1753: 121-127. J Qiu, N Yamada, J Tani, et al. Fabrication of piezoelectric fibers with metal core. Pmc. of SP IE's 10th International Symposium on Smart Structures and Materials. San Diego, CA. , Active Materials: Behavior and Mechanics. DC Lagoudas, Ed., 2003, 5053: 175-183. G Sebald, J H Qiu, D Guyomar. Modeling the lateral resonance mode of piezoelectric fibers with metal core. Journal of Physics D, 2005, 38: 3733-3710. Qian Li, Ying Yang, Dandan Wan, et al. Microstructural characteristics and electrical properties of x Pb(Mg 1n Ta'/3)O,-(0. 1-.T)Pb(Mnl/3Sb2/')O,-0. 9Pb(Zr0.5zTio.48)03 high power piezoelectric ceramics. Materials Science and Engineering B, 2009, 163: 115-150. (in Chinese) J Ryu, D SPark, D Y Jeong. Effect of LazO, doping on the piezoelectric properties of PbZr03-PbTi03-Pb(Zn1l3 :'-Ib2/3) 03-Pb(Snl/3 Nbz/3) 03-yMn03 ceramics for high-power applications. Journal of Ceramic Processing Research, 2009, 10: 386-390. (in Chinese) G H Hacrtling. Ferroelectric ceramics: history and technology. Journal of the American Ceramic Society, 1999, 82: 797-818. (in Chinese) Fuxue Zhang, Likun Wang. Modern Piezoelectricity (Volume 1, Second Edition). Beijing: Science Press, 2003: 97-98. (in Chinese) J Van Randeraat, R B Setterington. Piezoelectric Ceramics. Mullard Limited, 1971: 15-16. S E Park, T R Shrout. Ultrahigh strain and piezoelectric behavior in relaxor based ferroelectric single crystals. Journal of Applied Physics, 1997, 82: 1801-1811. G Roger, D Busch. A survey of micro-actuator technologies for future spacecraft missions. [2009-05-26]. http://www.robotstore.eom/support. asp. S E Park, W Hackenberger. High performance single crystal piezoeleetries: applications and issues. Current Opinion in Solid State &. Materials Science, 2002, 6: 11-18. M Levy, S Ghimire, A K Bandyopadhyay, et al. PZ:'-I-PT single-crystal thin film monomorph actuator. Ferroelectrics Letters Section, 2002, 29 (3-4): 29-40. K S Moon, M Levy, Y K Hong, et al. Axial displacement measurement of a single-crystal actuator using phase-shift interferometry. IEEE Transactions on Industrial Electronics, 2005, 52 (4): 953-959. M Yang, M Zhu, C Robert, et al. Design and evaluation of linear ultrasonic motors for a cardiac compression assist device. Sensors and Actuators A, 2005, 119: 214-220. S Dong, L Yan, :'-I Wang. A small, linear, piezoelectric ultrasonic eryomotor. Applied Physics Letters, 86: 200505350l. Z Y Feng, T H He, H Q Xu, et al. High eleetrie-field-indueed strain of Pb(Mg 1/ 3Nbz/3)O,PbTi0 3 crystals in multilayer actuators. Solid State Communications, 2001, 130 (8): 557-562.
[31J [32J
S E Park, T R Shrout. Relaxor based ferroelectric single crystals for electro-mechanical actuators. Materials Research Innovations, 1997, 1 (1): 20-25. S Genti, D Damjanovie, :'-I Setter. Pb(Mg 1/3Nbz/3)O, and (1-x) Pb(Mg1n:'-lb'n)O,-.T PbTi03 relaxor ferroelectric thick films: processing and electrical characterization. Journal of Electroceramics, 2004,12 (3): 151-16l.
[:l3J
V Y Topolov. Orientation relationships between electromechanical properties of monoelinic
Chapter 2
Fundamentals of Piezoelectricity and Piezoelectric'"
O. 91Pb (Znl!' Nbz/,) 0,-0. 09PbTi0 3 single crystals.
[31J
[35J [36J [37J
49
Sensors and Actuators A-Physical.
2005.121 (1):118-155. S C Woody. S T Smith. X N Jiang, ct al. Pcrformancc of single-crystal Pb(Mgl/ 3 Nb z/3 ()3)32%PbTi0 3 stacked actuators with application to adaptive structures. Review of Scientific Instruments, 2005, 76 (7): 075112(1-8). T Takenaka, H :'-Iagata. Current status and prospects of lead-free piezoelectric ccramics, J. Euro. Ceramic. Society, 2005, 25: 2693-2700. Zhiwen Yin. Physics of Dielectrics (Second Edition). Beijing: Science Press, 2005: 778-831. C P Araujo, J F Scott, G W Taylor. Ferroelectric Thin Films: Synthesis and Basic Properties. Amsterdam: Gordon and Breach Scicnce Publishcrs, 1996: 1-8.
Chapter 3
Fundamentals of Tribology and Tribomaterials for Ultrasonic Motors Tribology is defined as "the sCIence and technology of phenomena occurnng at the contact interface between objects", and its main topics are friction and wear of materials. Ultrasonic motor is a new tribological actuator, which uses the friction at a contact area between a stator and rotor to convert the ultrasonic vibration of the stator into the linear or rotational motion of the rotor. It is evident that the ultrasonic motor with friction drive possesses features such as self-lock without power and a high self-lock torque. As far as the locking property of USM is concerned, the self-lock torque is higher than its stall torque, while the rotor's inertia is low. This indicates that the ultrasonic motor has rapid (millisecond scale) response property. Because USM transmits the power via friction at the contact area between the stator and rotor, stable and relatively high friction force at the contact interface is required. Since sliding wear between the rotor and stator is inevitable,
high wear resistance of tribomaterials in USMs (stator and rotor) is then essential to maintain precision control, because their wear causes changes in the contact condition between the stator and rotor and leads to a decrease in control accuracy. Generally, friction characteristics include output power property of friction surfaces, the microstructure of wear surface and the tribological property at contact surfaces, whereas the phys-chemical properties of tribomaterials include an elastic modulus, wear resistance, anisotropy, dependence to environments, etc.: 1 RJ Therefore, how to match the friction and wear characteristics between stator and rotor pairs is a key to guaranteeing the performance of USMs. Currently, the tribological behaviors of tribosystems without vibrations have been investigated. If the vertical and tangential high-frequency vibration components along .1':, y, and z axial directions are superimposed on the tribosystem, the friction and wear behaviors become sophisticated. Thus, it is imperative to study the friction and wear behaviors of tribomaterials under ultrasonic vibration. To improve the reliability and stability of USM, an advanced functional surface technology and nanotechnology must be used to adjust and enhance the tribological properties between the stator and rotor in adverse circumstances. Furthermore, it is important to devclop experimental methods for estimating tribomaterial life and efficiency because the life and the efficiency of USM largely depend on the tribological properties of the stator and rotor pairs. It is obvious
C. Zhao, Ultrasonic Motors © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011
Chapter 3
Fundamentals of Tribology and Tribomaterials···
51
that friction interfaces for different ultrasonic motors exhibit various action fashions. However. due to the rotor or mover being impelled to move according to the elliptical motion at the contact points between the stator and rotor. the influences of the different action fashions on the stator and rotor are identical[9-15]. Therefore. it is of vital importance to analyze the tribological effect of the stator and rotor pair if we want to prolong the life of USM. In this chapter. the basic tribology will be first introduced. and then the preparation of tribomaterials in USM will be addressed. The influence of the phys-ehemieal properties of the tribomaterials on tribologic characteristics will be analyzed. After that. the experimental methods to determine the tribomaterial performance will be discussed.
3. 1 3. 1. 1
Basic Tribology Surface of Tribomaterials
In a viewpoint of tribology. the friction between two sliding surfaces is largely governed by physical conditions and chemical interactions between sliding surfaces and environments. Physically. rough surfaces could create higher friction coefficient. while chemical interactions between two sliding surfaces and the environment can play an important role in the friction and wear behavior of tribocouples in different environments. Usually. the practical solid surfaces arc formed by cutting. grinding and polishing. These surfaces look smooth and arc sometimes called mirrors. But even a mirror surface is still rough microscopically. Since the solid surface has microasperities. the surface roughness is estimated by peaks and valleys with various amplitude and space. These microasperities with long wavelengths on the surface arc formed owing to the vibration of the work piece or tools during processing. The micro asperities arc distributed directionally or isotropically on the surface depending on processing methods. When the solid surfaces are processed via turning. milling and planning. the asperities arc distributed directionally. whereas the asperities arc distributed isotropic ally or equiprobably as the surfaces arc processed via electropolishing and lapping. The surface roughness is determined by R, (centerline average roughness). Ry (average roughness at ten points) andR,(maximum altitude roughness). respectively. Normally. R, is more often used to show the roughness value of surfaces. For the stator and the rotor in USM. their surface roughness R, is lower than 0. 51l-m. In order to measure the roughness value for such solid surface. a surface profilometer is popularly used. and the cross-sectional profile of the surface is measured as shown in Fig. 3. 1. Furthermore. the surface microstructure of metal will be changed during cutting process. The surface of metals usually consists of several layers which arc formed during their machining processes[16] • as shown in Fig. 3. 2. From the metal matrix. the first layer is the deformation layer(also known as the strain hardening layer) with about l0ll-m thickness. and then the Bielby layer is formed on the deformation layer due to the surface melting. flowing and quenching. Thus. the microstructure of Bielby layer is amorphous or microcrystal. The oxide layer is formed by the chemical reaction between metal
52
Ultrasonic Motors Technologies and Ap plicalions
surface and oxygen in air. The outside layer is the absorption layer or contamination layer formed by the absorption of gas or liquid polar molecule on the solid surface in environments. It is obvious that the surface roughness and microstructure of tribomaterials have more influenees on the tribologieal properties of tribomaterials.
o
o
Dusl panicle wilh dia meter of I ~m Oxide laye r
-I
0. I
0.2 0 .3 0.4 0.5
0.6 0 .7
0 .8
x/ mm
Fig. 3. 1
3. 1. 2
Surfacc morphology of a stator
Fig. 3. 2
Typical surface of metal
Friction and Its Classification
The importance of friction may be seen in daily life. To decrease energy consumption in overcoming friction during sliding, the reduction of friction is extremcly important in modern technologies. However, when people walk or car moves. they need sufficiently high friction to push their body to move forward. Thus, it is imperative to control friction in our modern life. If friction effect exits on the contact surface between two sliding objects. which are called as a tribopair. Fig. 3. 3 shows a tribosystem constituted by objects A and B. It is clear that the object A is pressed firmly on the object B at a normal load of P. When object A is pushed by an external force F. a rclative motion or motion trend occurs at the contact surface between A and B. At this moment, A phenomenon to impede the rclative motion appears at the contact area.
This phenomenon is
called as friction. The force impeding the object motion at the contact surface is called as friction force, marked as Fr. The magnitude of Fr is related to the normal load. the contact surface status and the tribopairs.
Fig. 3. 3
Tribosystem
The following Coulomb's friction law:]6] is the most classical equation to
Chapter 3
Fundamentals of Tribology and Tribomaterials···
53
describe the above tribosystem (3.1)
where /1 is a friction coefficient. In the viewpoint of the macro-motion state, frictions can be classified into static friction and dynamic friction. As seen in Fig. 3. 3, thc two objects A and B arc initially kcpt in relativcly stationary statc at thc normal prcssure P. When thc extcrnal force F is zcro, therc is no friction between A and B. When F increases gradually in a certain rang, the relative micro-motion will occur between A and B. In this case, the friction force Fr is callcd as a static friction force, which corresponds to a static friction cocfficicnt /1,. When F is higher than the maximal friction forcc, the rela ti ve macro-movc bctwccn A and B will occur. Thcn this friction force is called as a dynamic friction force, which corrcspond to a dynamic friction cocfficicnt /1d. According to the relative motion of the tribopair, the friction is classified as sliding friction, rolling friction, rolling and sliding friction: (1) Sliding friction is thc friction state that thc relativc spccd on thc contact surfacc of the tribopairs are not cqual to zcro. (2) Rolling friction is thc friction statc that the relativc specd at somc actual contact points are zero. (3) Rolling and sliding friction is the friction state that combines sliding friction and rolling friction. According to thc friction states of thc contact surface of tribopairs, thc friction can be classified into dry, boundary, fluid and mixed frictions: (1) Dry friction is the friction condition that there is no lubricant between two objccts; usually the object surfacc absorbs gas, aqucous vapors and so on. (2) Boundary friction is thc friction condition that thc cxtremely thin film of the lubrication oil separates the sliding surfaces of two objects. (3) Fluid friction is the friction condition of which the surface of two objects is completely separated by fluid film and the friction characteristics is decided by the fluid viscosity. (1) Mixed friction is the mixed conditions including dry friction, boundary friction and fluid friction.
3. 1. 3
Friction Mechanism
To cxplain thc occurrcncc of friction, Amontons put forward the mechanical junction thcory in 1699, while Tomlinson and Hardy put forward the molecular attraction theory in 1929 and 1936, respectively. Since 1938, Bowden had done detailed works about tribology. For example, Bowden and Tabor have distinguished the large difference bctwcen cffectivc contact arca and true contact arca in 1942. Finally, the adhcsion thcory had becn propos cd by Bowdcn in 1950 to elucidate the friction mechanism. Now, the adhesion theory and its development are reviewed here- 16 -.
1. Simple adhesion theory Even in the case of a mirror surface, the surface displays rough asperities
54
Ultrasonic Motors Technologies and Ap plicalions
microscopically. When one solid contacts with the other, the surface mlcroasperisties will weld together due to the adhesive between a tribopair. When one of the solids moves on the other, the microweld junctions will break. So the entire friction process changes from the adhesion of asperities to the shear fracture of junctions alternatively. At the normal load pressure of p, the contact stresses at some contact asperities are so high that the plastic deformation occurs at contact zone, and then the contact area increases to the all area bearing load P. Suppose the yield strength of an ideal elastic-plastic material is rr, and the true contact area between two solids is S" then it has P=rr,S,. Once an object slides relatively against the other, the adhesive junctions arc fractured via shearing. This indicates that the shear force of the junctions is the main part of the friction force F. So the friction Fr force can be expressed as FI
S,T;,
=
=
(Plrr,)r;,
(3. 2)
and the dynamic friction coefficient (friction coefficient) is calculated as fl.d
=
FrlP
=
T:/rr,
(3. 3)
where T;' is the shear strength of the adhesive junction. It is obvious from Eq. (3. 3) that the friction coefficient is independent of the effective contact area, but is directly proportional to the shear strength of adhesive junction and inversely proportional to the yield strength of the ideal elastic-plastic material. If we consider the strain-hardening effect of tribomaterials, the shear strength Tb of the softer material is used to replace T;, in Eq. (3.3). Then (3. 1)
according to Eq. (3. 4), the friction coefficients of o. 2 for most metals arc acquired, but experimental results are mostly between O. 2-0. 5. This difference indicates that the simple adhesion theory has to be modified L16_
2. Modified adhesion theory In the situation of static friction, the true contact area S, is directly proportional to the normal load P. When the two objects in the tribosystem slide relatively, the true contact area S, will increase. Assuming the adhesive junction's yield is related to the composite stress of the compressive stress given by the normal load P and the shear stress given by tangential force, as shown in Fig. 3. 1, an empIrical equation for the modified model is introduced as (3. 5)
where a is an experimental constant. When a natural contamination film is formed on the contact surface and its shear strength is TI' there is (3. 6)
where Tb is the shear strength of the softer material in tribosystem. f3 is less than 1. When the ratio of Fr I S, is lower than Tr' with P increasing the contact area increases, while the contact area stops increasing as the ratio of FilS, equals to Tr. If the adhesive junction is sheared, the tribopairs start to slide over each other. If T in Eq. (3. 5) is replaced by Tr' the sliding criterion of the tribopair can be
Chapter 3
Fundamentals of Tribology and Tribomaterials···
55
s,
Fig. 3. 4
Compressive stress and shear stress at junction
expressed as (3.7) it is known from Eq. (3. 7) that the number of adhesive junctions will increase if F is very high. Then aT~ ""=' a; or a = a; / T~' so a; = ari / f3 2. Combining those with Eq. (3. 7), the friction coefficient can be derived as /1d
f3 /aCl-f32)
(3. 8)
If the contact surfaces are well cleaned and the contact interface has good adhesion, Tr is close to a, of the softer material in contact and f3 is close to unit. It is clear from Eq. (3. 8) that /1d becomes infinite when f3 is equal to 1. When f3 decreases from one, /1d decreases quickly. Due to the shear strength of the contamination film lower than that of metal and the cease of junction growth, f3 is close to zero. Thus, Eq. (3. 8) can be represented as /1d = Tr / a" which agrees with the simple adhesive theory. Although the modified adhesion theory can explain the tribologieal phenomena of metals, it has been criticized because of the following questions such as: CDthe agreement between the theoretical calculation and the experimental results of the friction coefficient is not good; (2)the effect of surface roughness on friction is not considered in this theory; Gil there is a lack of evidence that strength is necessary for the junction formed. To overcome the shortcomings of the adhesion theory, Kragclskii proposed the molecular-mechanical theory based on the adhesion theory and the molecule attraction theory in 1939. Under very high pressure, the mieroasperities on the real contact surface for a tribopair arc mutual chimerism, and the micro asperities of the harder object arc impressed into the softer one. Moreover, the molecular attraction force is existed at the contact zone. Because the motion process is to overcome the mechanical chimerism of mieroasperities, ploughing and the molecular attraction force in the tribosystem, thus, the friction force is a sum of all tangential stresses induced by the mechanical chimerism of mieroasperities, ploughing and the molecule attraction of the contact junction, and expressed as (3. 9)
Ultrasonic Motors Technologies and Ap plicalions
56
where a and yare related to phys-mechanical properties of contact surface. Combining Eqs.(3. 9) and (3. 1). the friction coefficient can be derived as
y+ aS,1 P
{1d =
(3. 10)
where y is the constant friction coefficient obtained from the mechanical chimerism theory. while ,I P is the variable of y after considering the influence of molecular attraction. This theory considers each factor comprehensively. and is usef ul not only to elucidate the mechanism of dry friction and boundary friction. but also to explain the tribology of metal and polymer material. The experimental friction coefficients arc listed in Table 3. 1. which shows the static and dynamic friction coefficients of tribomaterials with smooth surface.
as
Table 3. 1
Friction coefficient for normal tribopairs Static friction coefficients
Dynamic friction coefficients
Materials "fo lubricant
0.15
Steel-steel
o.
Lubricant
No lubricant
10-0. 12
o. o. o. o.
Steel-soft steel Steel-cast iron
0.30
Steel-bronze
0.15
Sol t steel-cast iron
0.20
Sol t steel-bronze
0.20
o.
18
Cast iron-bronze
o.
Bronze-bronze
Ebonite-steel
o.
0.05-0.10
20
o.
18
0.05-0.15
15
0.05-0.15
10
o. o. o. o.
18
0.07-0. 15
15
0.07-0. 12
15-0.20
0.07-0. 15
20
0.07-0. 10
10
0.01
Pure aluminum-brass hardened
o. o. o.
Steel-polycarbonate hardened
0.30
Steel-polyformaldehyde powder
0.16
Metallurgy-steel powder
o. 10 o. 10
Pure aluminum-steel
Bronze-bakelite
Metallurgy-cast iron
3.1.4
10-0.20
0.05-0. 15
o.
Cast iron-cast iron
10-0. 15
Lubricant
15
17 24 27
Wear Mechanism 1l718 -
Wear is the successive removal of surface materials by repeated friction and is mainly caused by microscopic mechanical fracture. Even when surface has some chemical reaction products. such as oxides. the volume loss from surface occurs mechanically in many cases. Although the various wear mechanisms have been proposed. it is difficult to predict the wear loss. It is elear that the wear process involves fatigue. fracture. corrosion and plastic deformation of tribomaterials. The wear mechanism is elassified by Burwell. Main wear mechanisms arc ad-
Chapter 3
Fundamentals of Tribology and Tribomatcrials···
57
hesive wear, abrasive wear, fatigue wear and corrosIve wear, respectively, which arc elucidated as follows.
1. Adhesive wear Adhesive wear is a form of wear which occurs when two smooth surfaces arc slid against each other, and the fragments arc pulled off from one surface to adhere to the other. Adhesive wear always arises from the formation and shear fracture of the junction. When the adhesive junction strength is lower than that of tribomaterials, shear fracture occurs at the joint interface, and the transfer of material is not obvious and wear rate is low. When the adhesive strength is higher than the yield strength of softer material in tribosystem, fracture takes place in the subsurface of softer metal ncar joint, and then wear become mild. When the junction strength is higher than those of tribomaterials, shear failure mainly occurs in the subsurface of soft metal. The fragments adhered to the hard metal make the softer surface scratched. If the junction strength is much higher than the shear strength of tribomaterials, shear fracture occurs at the deeper position of one or two metals, and then wear become severe. 2. Abrasive wear Abrasive wear is the form of wear which occurs when a rough hard surface slides on a softer surface, and ploughs a series grooves. The material originally in the grooves is normally removed in the form of loose fragments. Abrasive wear can also arise when hard and abrasive particles are introduced between sliding surface. In this situation, the abrasive grains adhere temporarily to one of the sliding surfaces, or else arc embedded into it, and plow out grooves in the other. The form of wear is generally called as the three-body abrasive wear. Usually there is the extremely high stress at contact area between abrasive grain and sliding surfaces, which makes the tribomaterials deform plastically and fatigue or fragment. If abrasive wear is caused by hard and rough surface, the form of wear is referred to as the two-body abrasive wear. When the motion direction of particles is parallel to the solid surface, the stress at contact zone between particle and smooth surface is low, which is characterized by the scratch line and shallow grooves on the surface. If the motion direction of particles is vertical to the solid surface, the collision contact stress at interface between particles and surfaces is high, which is characterized by the deeper groove on the surface and large size particles peeled off.
3. Fatigue wear Fatigue wear is observed during repeated sliding or rolling over a track. The repeated loading-unloading cyeles may induce the formation of surface or subsurface cracks, which eventually results in the break-up of the surface with the formation of large fragments, leaving large pits in the surface. Fatigue failure depends on the amplitude of the cycle shear stress. If the shear stress exceeds the endurance strength of materials during rolling, the wear particles arc generated by the initiation and propagation of crack. For the rolling contact, cracks arc usually initiated in subsurface. If the contact condition is the mixture of rolling
58
Ultrasonic Motors Technologies and Ap plicalions
and a little sliding, the damage will occur close to surface. 4. Corrosive wear The mechanical-chemical reaction occurs at a sliding contact zone in the corrosive environment. and the corrosion elements are observed on the sliding friction surface. During sliding friction, the corrosion clements on the sliding surface arc worn away so that the corrosive attack can continue. This indicates that the corrosion and friction are promoted mutually in corrosive wear.
3. 1. 5
Wear Surface for Stator and Rotor of Ultrasonic Motors
Generally, wear occurs as a result of friction. For TRUM-60, its rotor surface is covered with the tribomaterial, and its stator is made of copper. Fig. 3. S shows the microstructure of wear track on a stator and rotor. As seen in Fig. 3. 5. it is clear that the plough grooves are formed on the worn surface owing to adding hard minerals such as alumina into the friction material as reinforced phase. Fig. 3. S (a) shows the optical microscope of a copper stator. The plough grooves arc generated on the worn surface of the stator owing to friction, and their direction is identical to the rotation direction of the rotor. If the hardness of tribomaterials (base materials) was lower than that of substrate metal, the plough grooves are formed on the rotor's surface. Thus, the wear mechanism is abrasive wear. It is obvious from Fig. 3. 5 (b) that the plough grooves arc formed on the rotor's surface owing to the friction effect, and the grooves' direction is the same as that of the rotor's rotation. Fig. 3. S(c) shows SEM image of polytetrafluoroethylene-based tribomaterials for rotor. As seen in Fig. 3. S (c), besides the plough grooves. the fragments arc observed to be pulled off the surface. This indicates that the wear mechanism is the mix wear including abrasive wear, fatigue wear and adhesive wear for the tribomaterials in USM.
(a) 0pl icalmi eroscopic ofstalor
(b) 0pliealmieroscopie ofrolor
Fig. 3. 5
3.2 3. 2. 1
(e) Scanning e el ctron microscopic ofrolor
Microscopic image of wear track
Tribomaterials Used for Ultrasonic Motors Basic Requirement, Classification and Selection Principle
In order to increase the mechanical characteristics and running life of ultrasonic
Chapter 3
Fundamentals of Tribology and Tribomaterials···
59
motor, the surface of a stator or rotor is usually coated with tribomaterial or modificd using othcr surface processing methods. Currcntly, therc arc two elcments in the criteria to evaluate the ultrasonic motor's performances: CDthe ultrasonic motor posscsscs thc good output pcrformancc and running stability; @thc ultrasonic motor has the excellcnt reliability and running lifc. It is realized that the factors to influence the energy transmission are the situation of contact sur face (roughness, contact area), tangent friction force, longitudinal vibration velocity; but onc of main factors to induce thc ultrasonic motor running unstably and its life shortening is its tribology of its tribopairs.
1. Basic requirement and selection principle of tribomaterials in ultrasonic motors According to before-mentioned designing requirements, tribomaterials for USM should meet the following basic conditions: CD appropriatc static friction coefficient (0.15-0.3 for TRUM, greater than 0.2 for LUSM) , the coefficients of dynamic friction close to that of static friction, and thcre is no creep or crawl at low velocity; @good wcar-rcsistant pcrformancc and lcss wcar ratc for tribopairs surfaces; ®low frictional noise «15dB); @good surfaces hardness match of tribopairs; Ql) stablc phys-chcmical propcrties at room, high and low tempcraturc, low or high temperature tolerance; ® good vibration resistance and impact resistance propertics. The selection of friction coefficient will depend on the designing requirement of ultrasonic motors. For the motor of short-stroke, discontinuous working and short lifc, it is suitable to selcct thc tribomaterials with high cocfficicnts. But for the motor of long-stroke, continuous working and long life, it is suitable to select the tribomaterials with lower coefficients. Furthcrmore, thc tribomaterials' lifc is one of thc important factors deciding the running-life of ultrasonic motors, so it is especially important for the tribomaterials to have high wear-resistance ability. Due to the generation of frictional hcat, tcmpcraturc on thc friction surface will incrcasc with the running timc, and finally approach to a balance temperature higher than 100'C. Thus, it is more important to guarantee ultrasonic motors running stably if the tribomaterials havc good tcmperature stability of phys-chemical propcrties. 2. Classification of tribomaterials Tribomaterials for USM are often composites, which consist of matrix, relllforced filler and friction regulator. Matrix is used to form the main-body of tribomaterials, whilc rcinforced fillcr is used to cnhancc the mechanical charactcristics, and friction regulator is used to regulate the friction coefficients so as to enhance thc output torque and cfficicncy of USM. Morcovcr metal coating is also uscd as the tribomatcrials of TRUM to keep it running discontinuously. For cxample, the elinvar coating is used in the TRUM of camera. For thc tribomaterials uscd in USM, wc must consider how to match thcir friction cocfficient and wear rcsistance. Although the addition of hard assist materials could enhance the hardness of tribomaterial, the high content of hard assist materials leads to thc mating pair's surface becoming worn. If a little of
60
Ultrasonic Motors Technologies and Ap plicalions
friction regulator is added into the matrix, the wear resistance and stability of tribomaterials would be enhanced. Then no stick-slip occurs, and vibration and noisy will be reduced. Based on the above-mentioned guiding theory, the tribomaterials used for USM could be classified as: CD polymer matrix; @ ceramic coatings; ® powder metallurgy; @metal coatings. There arc many kinds of tribomaterials for USM in the world. In Japan, Endo and Sasakp9: have reported a tribomaterial mainly made of neoprene, while in Germany, Rehbein and Wallasehek[2°:have devcloped a PTFE-based tribomaterial consisting of PTFE, polyimide, carbon fiber and steel. And in China, according to adhesive method, Baoku rjC21: has developed a tribomaterial with the main components of bisphenol type epoxy resin, phenol-formaldehyde epoxy resin, modified imidazole curing, frictional coefficient regulator, KH500 silane coupling agent and hardness regulator. Xuejun Liu, Tongsheng Li, et aI L22 - have developed an aromatic polyamide-based tribomaterial, and its main components are aromatic polyamide, cuprous chloride, graphite and carbon fiber. lianjun QUL23-21_ has reported a tribomaterials with the modified PTFE or nano PTFE. Recently, Zhiyuan Yao, Qingjun Ding and the author"-26 J have developed a series of tribomaterials used for the rotor and the stator. For coatings used in USM, Seok-Jin Yoon in Korea has indicated that the TiAI)J, Ti)J, DLC and Si-DLC coatings could be used on the stator- 27 -.
3. 2. 2
Influence of Composition on Tribological Properties
For TRUM, the matrix materials arc epoxy resin, phenol-formaldehyde reSin, PTFE, polyimide, neoprene, acrylonitrile-butadiene rubber (NBR) , etc., while reinforced fillers are aramid fiber, carbon fiber, wear resistance powder (mineral), wollastonite, calcium carbonate, alumina, etc. The proper addition of reinforced fiber will change the elastic modulus and the anisotropy of materials, and increase frictional coefficient as well as enhance the locked torque of motors. Due to the aramid fiber having high tensile strength as well as good thermal resistance and its friction coefficient higher than carbon fiber, so that aramid fiber is an ideal reinforced fiber. If the anti-wear powder is added into the tribomaterials made of phenol-formaldehyde resin modified by nitrile-butadiene rubber, the tribomaterials exhibit the high friction coefficient and then the locked torque of ultrasonic motor increases:"]. However, after the ultrasonic motor run for some time, its locked torque will decrease. This indicated that the frictional properties of the stator and rotor tribopair are not stable and the anti-wear ability of the stator/rotor tribopair is poor. In order to improve the anti-wear property of the material, the frictional regulator is added into the tribomaterial. At present, frictional regulators arc PTFE, copper oxide, molybdenum disulfide, graphite and copper powder etc., which can adjust the frictional coefficient. They arc absorbed on the surface of the anti-wear powder and distributed into the soft adhesive matrix to form the particle with specific function, and then could regulate the macroscopical friction property of tribomaterials. Table 3. 2 shows related properties of
Chapter 3
Fundamentals of Tribology and Tribomaterials'"
61
some raw materials [or tribomaterials based on polymers. Table 3. 2 Related properties of some raw materials for tribomaterials based on polymers Raw materials
Related properties
Epoxy resin
Middle friction coefficient, brittle. good wear resist anee with fillers, bad temperature stability, high polarity, easy adbesion
Phenolic resin
Middle friction coefficient, brittle, middle wear resistanee with fillers, bad temperature stability, high polarity, easy adhesion
Polytetrafluoroethylene
perature stability, tiny surface tension, small com-
Low friction coefficient, high self-lubricity, good tempression modulus, friction regulator
Matrix
Polyimide
Chloroprene rubber
Butadiene-arylonitrile rubber
Polyphenyl ester Highdensity polyethylene Aramid fiber
Reinforcing filler
Carbon fiber
Alumina
Middle friction coefficient, brittle, good wear resist ance, good temperature stability
Good toughness, bad high temperature stability, good wear resistance with fillers, low efficiency
Good toughness, bad bigb temperature stability, good wear resistance with fillers, low efficiency
Tiny wear loss, tiny creep deformation, good radiation resistance, tiny injury to coupled part, brittle
Low friction coefficient, bad temperature stability
High tensile strength, high friction coeffeient, good temperature stability High tensile strength, low friction coeffeient, good radiation resistance, good temperature stability
high hardness, good wear resistance, good temperature stability
Molybdenum
Solid lubricant, low friction eoeffcient, high friction coefficient at high temperature
Graphite
Solid lubricant, low friction coefficient, good temperature stability, good chemistry stability, conducting
Friction regulator
Copper oxide
High friction coefficient, good temperature stability
Ceramic composites and metal coatings arc used for LUSM. Moreover ceramic composites arc mainly alumina-, titania-, chromium oxide-based composites, etc. As is known, alumina has high hardness, high brittleness, and low wear
62
Ultrasonic Motors Technologies and Ap plicalions
loss. If the alumina ceramics contains a certain amount titania, the alumina composites have good toughness, low wear loss and excellent heat insulation performance. The chromium oxide ceramics has low friction coefficient, good polishing performance and low wear loss. Recently, metal coatings are often used as tribomaterial for TRUM and LUSM, and their main components are nickel, chrome or their alloy. But short life is their shortcoming, so discontinuous condition is suitable for this technology. As above-mentioned, the composition of tribomaterials will have major influence on their tribological properties. The best composition of tribomaterial should be determined by using orthogonal matrix design with a few experiments.
3. 2. 3
Preparation of Tribomaterial
According to the type of tribomaterials, their preparation process method and the process equipment are different. The main equipment to prepare the epoxy resinbased tribomaterial is ordinary oven, while the main apparatuses to prepare PTFE-based tribomaterial arc hydraulic pressure machine bclow 20 ton and a high temperature sintering furnace above 100 'C. For the ceramic coatings, the main equipment is a plasma spraying device. Now, the preparations of the PTFE-based tribomaterial, epoxy resin-based tribomaterial and alumina-based tribomaterial arc introduced in detail here.
1. PTFE-based composite tribomaterial on rotor PTFE-based tribomaterials consist of the PTFE matrix, the reinforced agent of nano diamond powder and the regulator of copper powder. Its common compositions (molar percentage) are: CDPTFE matrix(60%-70%); @reinforcing agent 0%-25%); (3)regulator(5%-30%). It is clear that the matrix content is up to 60%. If the filler content is too high, the increase of hardness for tribomaterials will cause the abrasion wear of tribopair. There are three procedures to prepare the PTFE-based tribomaterial: CD three kinds of raw powders according to their molar ratios are mixed, stirred uniformly and dried up; @ the above mixture is filled into the mold, and pressed by the hydraulic machine to form an embryo of tribomaterial, then kept it at 40-60'C for 21-18h; C]the molding product is sintered at 370-380'C. As the temperature increases from 20 to 330'C, the heat speed is 40'C/h, while that is 30'C/h when the temperature rise from 330 to 380'C. When the temperature approaches to 380'C, it is kept for 4h. Figure 3. 6 shows the PTFE-based tribomaterial developed by PDLab. The PTFE-based tribomaterial is often adhered to a rotor. Firstly, the rotor is made via machining aluminum alloy. Then the sinter cd tribomaterial is cut into sheet (with the thickness of o. 2-0. 3mm) and adhibited on the rotor (as seen in Fig. 3. 7). 2. Epoxy resin-based tribomaterial on stator Epoxy resin-based tribomaterial consists of epoxy resin matrix, nano alumina reinforcing agent and PTFE regulator. Its common composition (molar percentage) are: CDmatrix of epoxy resin: 50%-60%; @reinforcing agent: 20%-35%;
Chapter 3
Fundamentals of Tribology and Tribomaterials'"
Tribomaterials based on polymers
Fig. 3. 6
Stators of TRUM with tribomaterials
Fig. 3. 8
63
Rotors with tribomaterials based on polymers
Fig. 3. 7
Rotors of BTRUM with tribomaterials
Fig. 3. 9
@regulator: 5%-30%. These raw materials including epoxy resin, PTFE, alumina, carbon fiber and curing agent arc cohered on the stator's surface and rotor's surface after being mixed up, and then lathe processes them to the required size after the composite is cured in heat at 80"C for 2 hours, as shown in Fig. 3. 8 and Fig. 3. 9. The frequency response experiments of stator arc performed with PSV-300F vibration measurement system using laser Doppler. A rotary tribometer designed by PDLab is used to test the wear and the friction of the samples. After running-in period of 10 hours, friction coefficients are tested under two different conditions: CD20"C, preload 100-250)J; @20"C ,prcload 100-250)J , voltage imposed on single face 15 V, frequency 37. 4kHz. The apparatus is located in a clean room with the relative humidity of 25%-50%. The rotor rotates at 12r/min, and the stator is fixed to the tribometer. Prior to testing, the eounterfaees are cleaned with ethanol, and dried. The normal load is continuously monitored and controlled with computer via using an eleetropneumatie valve. The data of normal load and friction force arc collected instantaneously. Table 3. 3 shows the friction coefficient of tribomaterials against the anodized aluminum rotor. It is obvious that the friction coefficient is not a constant value at different preload, and the higher the preload is, the higher the friction coefficient is. It indicates that the contact area increases with an increase of the preload due to the tribomaterial's toughness.
Ultrasonic Motors Technologies and Ap plicalions
64
Table 3.3
Friction coefficient between tribomaterial and aluminum rotor anodized Friction coefficient
Preload/)!
Ordinary state
Under ultrasonic vibration
Variation ratc/ %
100
o.
150 7
O. 119 4
-20.7
150
0.153 1
0.123 2
-19.7
180
0.151 3
0.131 6
-11.7
200
O. 158 1
O. 140 7
-11. 0
250
O. 160 7
O. 113 8
-10.5
The friction coefficient decreases 10 %-20 % under ultrasonic vibration in comparison to that of the ordinary state- 10J • Due to ultrasonic vibration and impact, the contact area between tribomaterial and rotor under ultrasonic vibration is less than that of the ordinary state, and the preload decreases, thus the friction force and the apparent friction coefficient all decrease. As seen in Table 3. 3, it is evident that the more the preload is, the less the decrease degree is. The rotor is impacted by the stator coated with tribomaterial. When the pre-pressure increases, the effects of impact to contact area makes reduce. This indicates that the decrease degree of the friction coefficient is little. Figure 3. 10 shows that SEM images of friction surface of epoxy resin-based tribomaterial on stator. There arc some microcracks after running for 200 hours, while there is obvious delamination after running for 600 hours. However, there are no ploughing grooves. It could be coneluded that the wear mechanism of epoxy resin composite are fatigue wear and adhesive wear. Therefore, the antifatigue performance and high cohesion energy density of the tribomaterials should be investigated when they are used in ultrasonic motors. In addition, the hardness, thermal stability and friction coefficient of the tribomaterials are usually considered as key factors of effecting or wear resistance of the tribomaterials.
(a) 200h
Fig. 3. 10
(b) 600h
SEM images of friction surface
Because the variation of temperature results in the migration of frequency response, thus, one of key factors affecting the stability and adjustability of ultra-
Chapter 3
Fundamentals of Tribology and Tribomaterials'"
65
sonic motor is the width of frequency response. With increasing the width of frequency response, the stability and adjustability of ultrasonic motor increase. Fig. 3. 11 shows frequency responses of the stators. It is obvious that there is no migration for the frequency response when the stator( Fig. 3. 11 (b)) is covered with epoxy resin composite and the half-peak width of frequency response increases about 2 times of that of the stator (Fig. 3.11 (a)) without the composite. This indicates that the stability and adjustability of ultrasonic motor increase significantly via coating tribomaterial on the stator.
rL 1 1 1 f L Ll1 20
20
30
30
40
40
50
FrequencylkHz
50
Trioomatcria ll iner
l(b'
Frequency/kHz
Fig. 3. 11
Frequency response for stator
The experimental results show that the half-power bandwidth of working mode responding curve is widened if the tribomaterial is adhered on the surface of stator (Fig. 3. 11 (b)). This causes the ultrasonic motor having more stable rotation speed. Besides PTFE type the other tribomaterials developed by PDLab can meet the requirement for all kinds of ultrasonic motors (Fig. 3. 9). 3. Alumina composite as friction material on stator of LUSM Compared with traditional electromagnetic motor, one of the advantages of ultrasonic motors is excellent transient property, mainly in rapid response, self-locking performance and precise positioning. At present, linear ultrasonic motors have been used in rapid response unit, high-grade instruments, and precision control devices. One of the key factors effecting on transient property is hardness of tribomaterial. Tribomaterials based on polymers will delay the response time because of the polymers' toughness and heat deformation. So inorganic composites are often used as tribomaterials in linear ultrasonic motors. Because it is difficult to bond inorganic composites to the stator of ultrasonic motor without any tackiness agent. The plasma spray process is basically the spraying of molten or heat softened material onto a surface to provide a coating. Material in the form of powder is inj ected into a very high temperature plasma flame, where it is rapidly heated and accelerated to a high velocity. The hot material impacts on the substrate surface and rapidly cools forming a coating. This
66
Ultrasonic Motors Technologies and Ap plicalions
plasma spray process carried out correctly is called a "cold process" (relative to the substrate material being coated) as the substrate temperature can be kept low during processing avoiding damage, metallurgical changes and distortion to the substrate material. Plasma spraying has the advantage that it can spray very high melting point materials such as refractory metals like tungsten, ceramics and zirconia unlike combustion processes. Plasma sprayed coatings arc generally much denser, stronger and cleaner than the other thermal spray processes with the exception of HVOF and detonation processes. Plasma spray coatings probably account for the widest range of thermal spray coatings and applications and makes this process the most versatile. The tribomaterial based on alumina is composition of the matrix of alumina and the regulator agent of titanium dioxide, etc. Its common composition (molar percentage) is: (1) matrix of alumina: 55%-80%; (2) the regulator agent of titanium dioxide: 10 %-10 %; (3) others: 0%-10%. Although in the prescription above, the ratio of matrix is up to 55%, however, the study shows that the ratio of the regulator agent of titanium dioxide has great effect on the friction properties of tribomaterial. It is clear that the high speed, torque, output efficiency and the efficiency of interface dynamical transmission would be acquired when adjusting the content of titanium dioxide in a certain range. The commercially available AI,O, powders with an average particle size of 40~70fLm arc used as a feedstock in the present study. The raw feedstock has the purity >99. Owt% of Al,0 3 component. And titanium dioxide powders with an average particle size of 50fLm are used as an additive to prepare the other feedstock of AI, 0, -TiO, composite. The composite powders with a content of around 20 % TiO, arc mechanically mixed in a rotary-vibrationmill, alcohol being used as a binder, and then suffered sieving and drying prior to the spraying.
Fig. 3. 12
AI, 0 3 - Ti0 2 tribomaterial
The DH-2080 atmospheric plasma spraying equipment made by Shanghai Dahao :'\Ianomaterials &. Thermal Spray Co., Ltd. in China is applied to prepare AI 2 0 3 -TiO, compositc coatings. Thc fcedstock powdcrs arc fed with a Twin-Systcm 1O-C. A mixturc of argon and hydrogen is uscd as plasma gas. During spra-
Chapter 3
Fundamentals of Tribology and Tribomaterials···
67
ymg. the substrates and coatings are cooled using compressed air. Stainless steel coupons arc uscd as substrates. Beforc spraying. the substratcs are dcgrcascd ultrasonically in acctonc and grit blasted with corundum. In addition. the plasma torch is utilized to spray powders onto the unheated quartz substrate in order to observe the spreading and flattening morphology of impacted droplets. Disadvantagcs of the plasma spray process are relativc high cost and complcxity of process.
3.3 3. 3. 1
Influence of Tribomaterials on Performance of USM Influence of Elastic Modulus and Hardness
Elastic modulus E and hardness H are two essential parameters of materials. Elastic modulus is relatcd to the material atom composition. whilc hardncss has relevancc to the organization structurc of materials. It is elcar that hcat trcatment has no influcncc on the elastic modulus. but has grcat cffect on hardness. espccially for mctal alloy. Gencrally. hardness depends on the local elastic-plastic deformation of solid material during indentation loading. and the elastic modulus can be calculated from unloading process. Based on the conventional depthsensing indentation method proposed by Oliver and Pharr. Chinese researchers dcrivcd an analytical relationship between the reduccd modulus and hardncss for solid materials. It is found that the hardness and thc elastic modulus are interrelated to each other through the recovery resistance of materials. Experimental results show two important features: CD the reduced modulus predicted by the new E'- H relationship is the same as that obtained by the conventional method; CZ) the elastic modulus and hardness determined by the simple set of procedures arc comparable to thosc obtaincd by using thc convcntional method: 2"J.
1. Influence of elastic modulus on USM The elastic modulus of tribomaterials is one of the major physical parameters determining the friction characteristics. The results show that the elastic modulus affects the no-load speed. output torque. output power and start-stop characteristic of ultrasonic motors. Thc variation of opcrating spccd with the elastic modulus is not simply lincar relationship. The prescnt theory :29J indicates that undcr the condition of no-load and thc certain prc-prcssure in the range(250-300N). it is available that thc contact area between a stator and rotor would decrease as the elastic modulus of material increases in the range( o. 2-1. 5GPa) • which induces the average tangential velocity of thc rotor increasing at thc contact arca. With thc incrcase of thc avcrage tangential velocity and the decrcase in elasticity sliding motion. thc noload speed of ultrasonic motors would incrcascs. If the elastic modulus excecds thc above-mcntioned rangc. the averagc speed at contact area increascs and thc elastic sliding decreases. However. the high elastic modulus makes the interface area between the stator and rotor decreased. which causes a friction drive force and cnergy convcrsion rate bcing lower. In this casc. thc no load speed of ultra-
68
Ultrasonic Motors Technologies and Ap plicalions
sonic motors decreases. The author29 J analyzes the influence of vanous elastic moduli on the output performance of ultrasonic motors using the simulation software for the traveling wave ultrasonic motor, and the results indicate that the contact width of the stator and rotor pair in a wavelcngth and the deformation strain of tribolaycr bccomc wide and large, with decreasing the elastic modulus of tribomaterials. On the contrary, the contact width between the stator and rotor will decrease with the increase of the elastic modulus or the contact stiffness. When the elastic modulus varies in thc rangc of o. 1-0. 5GPa, the no-load spced incrcases with mcreasmg the elastic modulus. The elastic modulus of tribomaterials also affects the locked torque and the output cfficicncy of USMs. According to thc contact models, if the tribolayer is soft and the contact area extend to the area beyond the points with the same circumferetial spccd of stator and rotor, the contact area includcs impcding area which weakcns thc stator's driving effect on rotor. Whcn the contact stiffncss of tribolayer is high, the contact arca would decrcase and becomc thc driving zone, and then thc locked torque incrcascs obviously. In meanwhilc, thc radial componcnt of thc contact forcc on this zone will bc low, and thc sliding loss on the intcrfacc will also decrcase. Thereforc, the output cfficiency of ultrasonic motors bccomcs high. From the above analysis, it is clcar that thc high rotational spccd, torque, output efficiency and thc cfficiency of dynamical transmission on thc intcrfacc would be acquircd whcn the elastic modulus of tribolayer properly incrcascs in a ccrtain rangc.
2. Influence of hardness The hardness of tribomaterials affects not only running speed and output torque, but also frictional noise. The influence of tribomaterials' hardness on the opcrating performance of somc linear ultrasonic motors has bcen rcported by Endo[19:. Thc author prepared several tribomatcrials, and studied the influence of hardness on the performance of USM. PTFE compositc, cpoxy composite, phcnolic composites, hard aluminum alloy, ccmented carbide as tribomatcrials on rotors are respectively paired to the stator made of phosphor bronze and piezoelectric ceramic. By controlling fillers and roughness, their friction coefficient can be adjusted to a similar valuc. Table 3. 4 shows the Vickcrs hardncss of fivc kinds of tribomaterials. It can bc seen that the rank of hardness is arranged from low to high: PTFE composite< epoxy composites < phenolic compositcs < anodizcd aluminum alloy < cemcnted carbidc. Table 3. 4 Tribomaterial
PTFE
Vickers hardness (HV)
11
Vickers hardness of tribomaterials Epoxy
Phenolic
Anodized
reSln
reSln
aluminum
Cemented carbide
39
80
453
1 120
Chapter 3
Fundamentals of Tribology and Tribomaterials···
69
Figure 3. 13 shows the curves o[ speed vs. torque [or the ultrasonic motor with five kinds of tribomaterials. It is evident that with an increase of the hardness of tribomaterials, the no-load speed increases, while the stalling torque decreases. Moreover, the difference o[ the no-load speed also decreases. In other words, the hard tribomaterials could be applied to the ultrasonic motors with high speed, while the soft tribomaterials could be used in the ultrasonic motors with high stalling torque. Because the ultrasonic motors with hard tribomaterials often run with high noise, thus polymer composites are a main kind o[ tribomaterials especially used [or traveling wave rotary ultrasonic motors.
220 200
"'-.-,
.. .:.......
' :":':-- ...."
180 160
'2 140 120 0::
"'!l" c.
Vl
, ....
~
Anodized al uminu m Cemented carbibe
..... . '.;: : ....
.~ ~ .
-E ."
.
PTF E Epoxy resin Phenolic resi n
100
.:- ....~~.... ,.. .... .
., ... . . .....
'" .. , ... ~
80 60 40 20 0 00
0_2
0.4
0.6
0_8
1.0
L2
IA
Output torqu (N -m)
Fig. 3. 13
3. 3. 2
Mechanical characteristics for an ultrasonic motor
Influence of Friction Coefficient
The output torque of ultrasonic motors will increase with the friction coefficient in a certain range. The increase o[ output torque will stop as the friction coefficient increases to a certain value. If the friction coefficient becomes higher. the torque could not increase obviously. In this situation. the wear rate of tribolayer becomes high, and the noise of ultrasonic motors become aloud. Thus, the running life o[ USMs becomes short. In the viewpoint o[ tribology. there are primarily two friction mechanisms: the first is the sliding resistance caused by the mechanical chimerism of asperities for tribopair. This is a mechanical component in friction force. The second is the shearing resistance caused by the adhesion [unction o[ molecules at contact area. It is a molecular component in friction force. In order to enhance the friction force of the stator and rotor pair at a certain pre-pressure, the friction coefficient of the stator and rotor tribopair should be high. The simple method to raise the friction coefficient is the addition o[ hard particles into the tribomaterials. The hard particles can increase the sliding resistance caused by the mechanical chimerism of asperities. When the surface with hard micro-asperities is pressed to the
70
Ultrasonic Motors Technologies and Ap plicalions
soft surface, the friction force is formed owmg to the ploughing resistance. In this casc, thcrc arc many ploughing groovcs and thc powdcr loss on thc surfacc of tribopairs in USM. The friction coefficient is increased using the second method, which decreases the surface roughness of tribopairs and augments the adhesion force between the molecules. In this case, the powder loss becomes slight. The above analysis indicates that the high-speed, high-torque, high-output power and interfacial dynamic transmission efficiency can be gained as the elastic modulus of the tribolayer increases in a certain range, and the output torque, efficiency, rotation speed, and power of ultrasonic motors can be enhanced by increasing the friction coefficient of tribomaterials.
3. 3. 3
Influence of Anisotropy
The tribolayer on stators or rotors with a certain thickness
IS
distributed on the
annular area. This area is r 2 ~ r~ r3 , as shown in Fig. 3. 14. The sand {} denote arc-length and angel respectively. The contact model between the stators and rotors of ultrasonic motors is very complicated (see Chap. 5). One simple model is that the tribolayers arc supposed as the axial and circumferential independent springs. If the elastic coefficients of axial and circumferential springs are k n and k., respectively, the dynamic friction coefficient is !1d and the deformation of the friction layer is 0, the pre-load of Po is equal to kno under static status, as shown in Fig. 3. 15. Tribomaterial
r
Fig. 3. 14 Rotor of traveling wanc USM
Fig. 3. 15 Deformation status of rotor and stator
If the tribomaterial is pasted on the rotor, the free surface of the tribolayer is against the surface of the stator. In the situation of the ultrasonic motor operating, the stator affects the rotor through tribolayer. Assuming that the axial(normal) pressure f.(r,{},t) and circumferential shear force f.(r,{},t) have influences on the rotor in the friction area, they arc respectively expressed as
f ( r, {} ,t) -- {kn (w + 0) , n
f, (r, {}, t)
0,
=
w+O>o w+o~O
sign(V" - V) !1dn (r,{), t)
(3. 11) (3. 12)
where w is the displacement (z direction) of points on the surface of the stator,
Chapter 3
Fundamentals of Tribology and Tribomaterials···
71
V" is the corresponding circumferential velocity, V, is the circumference speed at the contact point between the rotor and stator. In the above-mentioned model, k n and /1d have different effects on the interaction between stators and rotor. Eq. (3.11) shows that the value of k n influences the contact state of the stator and rotor pair. When thc stator and rotor contact mutually, Eq. (3. 11) is exprcsscd as fn(r,{),t)=knw+k,J'j, where the constant forcc of k,J'j is cqual to Po as prcload, while k n w is alternating force, which represents the interaction between the stator and rotor during operating, and
o. 5k
n
w' is the work done by alterna-
ting force in axial direction. When k n is high, k nwand o. 5k nw' become high. Due to ineffcctive work done by thc ultrasonic motor along axial dircction, the ultrasonic motor's energy would lose. When the stator and rotor contact mutually, Eq. (3. 12) is changed as f,(r,{),t)
=
sign(V" - V,)/1dkn (w+ 0)
(3. 13)
It is clcar from Eq. (3. 13) that the valuc of /1dkn affects the transmission of thc energy from the stator to the rotor. With an increase in the values of /1dkn' the tangential force between the stator and rotor increases at the proper pre-load. Actually, with a decrease in the value of kn' the deformation amount of the tribolayer increascs and the contact width bctween thc stator and rotor in a wavelength enlarges gradually. This indicated that the value of k n should vary in a suitable rangc. If Lt is a contact time in which the point G on the stator contacts with the rotor in one period, the work done by this point to the rotor is
(3. 14) wherc hi is thc distance from thc stator surfacc to thc neutral layer. It is indicated that the stator transmits the cffectivc encrgy to the rotor though thc tribolaycr along circumfcrential dircction. In herc, the tribolayer in circumfercntial direction is considered as the spring with the elastic coefficient of k" which decidcs the output cfficiency of thc ultrasonic motors. Thc output efficiency usually incrcascs with an increasc in thc value of k,. The above-mentioned analysis indicates that in order to increase the operating efficiency of ultrasonic motors, it is necessary to make the anisotropic tribomaterials with low vertical elastic modulus. But in order to obtain the high output torque, the friction coefficient and the circumferential elastic coefficient k, for tribomaterials should be high. The anisotropic tribomaterials prepared in this way are beneficial to improving the output characteristics of ultrasonic motors. Based on the preparation of isotropic tribomaterials, the anisotropic tribomaterials can be prepared increasing the circumferential elastic modulus k,. After glass or carbon fibers are added into the isotropic tribomaterials, the fibers are distributed and stirred circumferentially, and then the anisotropic tribomaterials arc acquired.
Ultrasonic Motors Technologies and Ap plicalions
72
3.4
Friction Testing for Tribomaterials
Currently, there are two methods to measure the friction coefficients of tribomaterials: The first method is to determine the traditional static friction coefficients. The second method is to measure the dynamic friction coefficients based on the operating principle of USM.
3. 4. 1
Quasi-static Friction Testing
1. Summary Quasi-static tribometer, as shown in Fig. 3. 16, is used to measure friction coefficients of tribomaterials at low speed. The friction coefficient measured by the method is called a quasi-static friction coefficient.
.. ~
Sensor
,-~!!!!!!~II'
Signal amplifier
Data acquisition card
Motion control card
Motion loading: vert ical, el vel , rotational;
Fig. 3. 16
Data acquisition : adhesive force, friction force(moment), friction coefficient
Schematic diagram of quasi-static tribometer
The tribometer is controlled by a computer, whose software system IS wmdows interface in Chinese, and operated easily. Its data analysis software can accomplish the data acquisition and storage, and translate the test data into Word, Excel or other general software. Data record system adopts 12bits AID converter, the record speed can reach to 1000kHz as the experimental curve is shown and the dynamic saving disk works. According to the configurations of different sensors, the tribometer can accomplish the adhesion, friction and wear experiments.
2. Operating principle The tribometer includes hardware and software systems. The hardware system consists of 5 parts: the level moving part, the vertical moving part, the rotation part, the force sensor, and the control box of the motor. Meanwhile the attachments to the tribometer include the motion control card of motors, the signal amplified card of sensors, the data acquisition card, computer, and so on. The motion compartments such as level motion, vertical motion and rotation parts all arc driven by step motors. Software system consists of the drive and control system of step motors, the data acquisition and the data analysis software.
Chapter 3
Fundamentals 01 Tribology and Tribomaterials···
73
As seen in Fig. 3. 16, the relative movement between tribopairs is generated via moving parts, and a ccrtain prc-prcssurc is imposcd to thc stator. Thc rcaltimc data collcction and storagc of prc-prcssurc and friction forcc arc carricd out by using sensor, signal amplifier, and data collection card. The control system includes computer, motion control card, and control program, and controls the motion dircction and spccd of motion parts and thc prc-prcssurc bctwccn tribopairs.
3.4.2
Dynamic Friction Testing
1. Summary The dynamic friction test is used to measure the dynamic friction coefficients of tribomaterials during friction. For the running USM, there are macroscopic and microscopic rclativc motion at thc contact arca of thc stator and rotor tribopairs simultancously. Thc microscopic relativc motion shows two aspccts: CD thc stators and rotors are in the contact state with pulsation variation, which makes the contact stress of the stator and rotor tribopair to change periodically; @there is altcrnating rclativc motion along circumfcrcntial dircction. This motion statc causcs thc intcraction bctwccn thc stators and rotors bcing complicatcd, and thcn thc uniquc friction charactcristics arc cxhibitcd. To analyzc friction cocfficicnt of tribomaterials during running, a dynamic friction test machine is made and provided by Harbin Institute of technology. This machine can simulate the motion at thc contact point of thc stator and rotor pair for ultrasonic motors, and thcn mcasurc thc dynamic friction cocfficicnts for ultrasonic motors. 2. Work mechanism Thc dynamic tribomctcr utilizcs thc front cnd of bar with longitudinal and flcxural vibration modes to simulate the elliptical motion of a surface point on a stator for ultrasonic motor. When the bar excites a composite ultrasonic vibration made of a longitudinal and a flcxural vibration, thcn thc bar front cnd producc a highfrcqucncy microscopic clliptical motion, and thcn thc clliptical functions arc simulated. The dynamic tribometer consists of mechanical system, signal output, and data collcction and transfcr systcm, ctc. Thc mcchanical systcm is a vcrtical structurc. It is convcnicnt for loading prc-prcssurc and thc amplificd output of thc instant friction driving force as the ultrasonic micro-tribo test is done. As seen in Fig. 3. 17, the tribometer includes the ultrasonic vibration parts to simulate the high-frcqucncy clliptical motion of thc point on thc stator for thc traveling wavc ultrasonic motors, thc prc-tightcning structurc to adjust thc prc-load and thc position of instant kinetic positive pressure sensor, the pre-tightening part to regulate the position of output and sensor, the supporting and position structures of transmission output axis and cxpcrimcntal tablc. Thc signal transfcr and output parts includc thc piczoelcctric scnsor to mcasurc thc instant dynamic driving force, and corresponding electric charge amplifier.
74
Ultrasonic Motors Technologies and Ap plicalions
IISp"cinl~~~z"" I~~tnicsen sor I sensor 2 Operation simulation equipment for TRUM
Fig. 3. 17
Data acquisition card
Computer
Charge amplifier
Schematic diagrams 01 dynamic tribomctcr
References [ 1
J
[ 2
[3
J J
[ 4
J
[ 5
J
[ 6
J
[ 7
J
[ 8
J
[ 9
J
[IOJ [llJ
[I2J
[13J
[14J
[15J
Shizhu Wen. Existing state and development of tribology research in China. Chinese Journal of Mechanical Engineering, 2004, 40 (11): 1-6. Zhongrong Zhou, Leo Vincent. Fretting Wear. Beijing: Scicncc Prcss, 2002. (in Chines c) T Ishii, S Ueha, K "Iakamura. Wear properties and life prediction of frictional material for ultrasonic motor. Japanese J oumal of Applied Physics, 1995, 34: 2765-2770. H Storck, W Littmann, J Wallasehek. The effect of friction reduction in presence of ultrasonic vibration and its relevance to traveling wave ultrasonic molors. [lltrasonics, 2002, 40: 379-383. T Yamaguchi, K Adachi, Y Ishimine, et a1. Wear mode control of drive tip of ultrasonic motor for prccision positioning. Wear, 2001, 256: 115-152. M Kurosawa. Efficiency of traveling wave type ultrasonic motors. J. Acoust. Soc. J pn, 1988,11(1): 10-16. N M Hagood, A J McFarland. Modeling of piezoelectric rotary ultrasonic motor. IEEE Trans. Ultrason., Ferroelee!., Freq. Contr., 1995, 42(2): 210-224. P Hagcdorn, T Sattel, D Spcziari, ct a1. The importance of motor flcxibility in traveling wave ultrasonic motors. Smart Mater. Struct., 1998, 7: 352-368. Hcming Sun, Chunshcng Zhao, Xiaodong Zhu. Simulation on friction characteristic of ultrasonic motor using longitudinal and torsional modc. Journal of Southeast University (Natural Science Edition), 2002, 32(1): 621-626. (in Chincse) Heming Sun, Hui Guo. Thc relation of preprcssure and output-torquc of longitudinal and torsional ultrasonic motor. Tribology, 2001, 21( 1): 52-54. (in Chinese) Hui Guo, Taizhc Tan, Xinbao "ling. Moving track of the surfacc particlc and torquc for thc ultrasonic motor using thc traveling wavc in the plane. Tribology, 2002, 22(5): 386-390. (in Chincse) Xiangdong Zhao, Changqing Liu, Hcming Sun, ct a1. Output characteristics of thc frictional interface of traveling wave type ultrasonic motors. Small & Special Machines, 2000, 21 (3): 21-22. (in Chinese) Xiangdong Zhao, Bo Chen, Chunsheng Zhao. "Ionlinearly frictional interface model of rotated traveling wave typc ultrasonic motor. Journal of Nanjing University of Aeronautics & Astronautics, 2003, 35(6): 629-633. (in Chinese) Hai Xu, Chunsheng Zhao. Contact process and friction analysis of linear ultrasonic motor. Journal of Nanjing University of Aeronautics & Astronautics, 2005, 37(2): 144-149. (in Chinese) Qianwci Chcn, Wciqing Huang, Chunshcng Zhao. Mcasurcmcnt of scrviec lifc of ultrasonic
Chapter 3
[l6J [17J [l8J [19J [20J [21J [22J [23J
[24J [25J
[26J
[27J
[28J
[29J
Fundamentals 01 Tribology and Tribomaterials'"
75
motors. Journal of Vibration, Measurement & Diagnosis, 2004, 24 (l): 19-22. (in Chinese) J Halling. Principles of Tribology. Beijing: China Machine Press, 1981. (in Chinese) Shizhu Wen. Principles of Tribology. Beijing: Tsinghua University Press, 1990. (in Chinese) Zhendong Dai, Min Wang, Qunji Xue. Introduction to the Thermodynamics of Friction Systems. Beijing: National Defense Industry Press, 2002. A Endo, N Sasaki. Investigation o[ [rietional material [or ultrasonic motor. Japanese Journal of Applied Physics, 1987, 26: 197-199. P Rhbein, J Wallasehek. Friction and wear behavior of polymer/steel and alumina/ alumina under high-fretting conditions. Wear, 1998, 216(2): 97-105. Baoku Li. Preparation [or new [rietion material. Technology on Adhesion & Sealing, 2001, 22(3): 7-8. (in Chinese) Xujun Liu, Tongsheng Li, Tian Nong, et al. Manufacture and application of aromatic polyamide based [rietional material. China Plastics Industry, 1999, 27(3): 25-26. (in Chinese) Jianjun Qu. Friction Driving Mechanism and Friction Material Research on Ultrasonic Motor. Dissertation for the Degree of Doctor of Philosophy. Harbin: Harbin Institute of Technology, 1998. (in Chinese) Jianjun Qu. The Contact Model and the Properties of the Friction Materials for Ultrasonic Motors. Post-doctoral Report. Beijing: Tsinghua University, 2001. (in Chinese) Zhiyuan Yao, Qingjun Ding, Chunsheng Zhao. Preparation and auxiliary tools of thermoset resin-based friction material and friction layer of ultrasonic motors. Chinese Invention Patent, CN200610040708. 5, 2006. Zhiyuan Yao, Qingjun Ding, Chunsheng Zhao. Using PTFE-based filled with carbon fiber as friction material of ultrasonic motors and its fabrication. Chinese Invention Patent, CN200610010709. X, 2006. H-P Ko, SKim, J-S Kim, et al. Wear and dynamic properties o[ piezoelectric ultrasonic motor with [rietional materials coated stator. Materials Chemistry and Physics, 2005 (90): 391395. Y W Bao, W Wang, Y C Zhou. Investigation of the relationship between elastic modulus and hardness based on depth-sensing indentation measurements. Acta Material, 2004, 52 (18) : 5397-5404. Chao Chen. The Research on Theory Model for the Rotary Driveling Wave Ultrasonic Motor. Dissertation for the Degree of Doctor of Philosophy. Nanjing: :'-Ianjing University of Aeronautics and Astronautics, 2005. (in Chinese)
Chapter 4
Fundamentals of Vibration for Ultrasonic Motors An ultrasonic motor is onc of thc most typical cxamplcs of utilizing vibration. In order to understand its motion mechanisms and design principles, it is necessary to start with clastic body vibrations. Vibration is a classic topic in mcchanical cngineering and many references can be found 11-2J. In this chapter the vibration of clastic bodics is discusscd. It providcs thc ncccssary thcorctical foundation for the subsequent chapters of this book and for readers who are interested in ultrasonic motor technologies, but have little exposure to mechanical vibration. In general, structures (including an ultrasonic motor's structure) are made of simplc componcnts such as bcams, platcs, and shclls. Thcy havc a continuous distribution of mass and stiffness, called a continuous system (elastic body), that has an infinitc numbcr of natural modcs (natural frcqucncics and corrcsponding mode shapes). Analytical solutions to an elastic body vibration equation arc limitcd to only simplc gcomctrics with spccific boundary conditions. In most other cases, numerical methods are used instead to obtain approximate solutions. Thc Finitc Elcmcnt Mcthod (FEM) is thc most cffcctivc onc, of which somc highly sophisticated software, such as NASTRA:'\J, A:'\JSYS, ATILA, etc., is bascd on. Most of thc numcrical analysis in this book wcrc donc by ANSYS, which is a powerful package capable of static and dynamic analysis, modal analysis, timc domain analysis of structurcs, ctc. Ultrasonic motors utilize the inverse piezoelectric effect of piezoelectric ceramic clcmcnts to gcncratc strcss or strain, which cxcitcs a stator (clastic body) to produce forced vibration response. The response is converted into the rotational or lincar motion of a rotor or slidcr by thc friction bctwccn thc stator and rotor. Thcreforc, in ordcr to dcsign ultrasonic motors, pcoplc must also mastcr thc forccd vibration of clastic bodics c, 4J.
4. 1
Natural Vibration of Elastic Body
In the section, we will introduce the natural vibration of elastic body, including bars (shafts, beams), plates, shells, etc. A straight elastic strut can undergo longitudinal, torsional, and latcral vibration. If x dcnotcs thc longitudinal (ccn-
C. Zhao, Ultrasonic Motors © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011
Chapter 1
Fundamentals of Vibration for Ultrasonic Motors
77
troidaD aXIs, and y and z represent the directions of principal axes of a cross section, the longitudinal vibrations take place in the x direction, the torsional vibrations occur about the x axis, and the lateral vibrations involve motion in either the
.Ly
plane or the
.LZ
plane. The strut subjected to longitudinal vibration is of-
ten called a bar. We consider first the longitudinal vibration of a uniform bar using a simple theory.
4, 1. 1
Longitudinal Vibration of Bars
In the condition of no external force and no damping, we can obtain the equation governing the natural vibration of the bar in the longitudinal direction:
psa2u at'
=
~(ESaU)
a.L
ax
(4. 1)
where u(x, t) is the displacement function of the bar in axial direction; S(x) , E(x) , and P(.L) are the cross section area, elastic modulus of material, and mass density of the bar, respectively. For a uniform bar, Eq. (4. 1) can be simplified as
au at' 2
E a'u
(1. 2)
p a.L'
The solution of the above equation can be obtained through the separation of variables. Assume that the solution can be expressed as (1. 3)
Substituting Eq. (4.3) into Eq. (4.2) and using the method of separation of variables can yield
d2Xt~t) d'KL) dx'
+w'q(t)
0
=
+ WE'I'·L '13...--I.( )
(1. 1)
0
(1. 5)
+ Bcoswt
(1. 6)
+ Dcosw J"f-.L
(1. 7)
=
From Eqs. (1.1) and (1.5), we can obtain
q(t)
~(.L)
=
=
Asinwt
Csinw J"f-.L
The complete solution of Eq. (1. 2) becomes
U(.L, t)
=
(Asinwt
+ Bcoswt) (Csinw J"f-.L + Dcosw J"f-.L)
(1. 8)
where w denotes the frequency of vibration, the function ~(.L) represents the mode shape, the constants C and D can be evaluated from the boundary conditions, the function q (t) indicates harmonic motion, and the constants A and B can be determined from the initial conditions of the bar. The general solution of Eq. (4.2) becomes
Ultrasonic Motors Technologies and Ap plicalions
78
11-]
According to boundary conditions of bars, we obtain the natural frequency of vibrationwn(n = 1,2,3,···) and corresponding modc shapes ~n(x) , which arc summarizcd in Tablc A. 1 and Fig. B. 1 in Appcndixes A and B, respcctively.
4. 1. 2
Characteristics of Natural Modes
All of ultrasonic motors make use of thc "mode" of elastic bodics. Thc word "mode" is used to describe either the natural mode of vibration ( W n ' ~n ) or the mode shape ~n. In other words, the nth mode refers to the nth natural frequency and corresponding mode shape, or refers only to the mode shape ~n. It has been noted that usc of word "modc" has becn very loosc in litcratures. From Tablc A. 1 and Fig. B. 1, it can be observcd that the mode is in fact a wavc in spacc whosc amplitude ratio of various points along axis direction of the uniform bar holds a constant for all time. The certain points (called nodes) on the bar undergo zero amplitude, whereas other points (called antinodes) attain maximum amplitude. The nodes and antinodes occur at regular spaces along the bar and remain the fixcd positions for all timc. This form of vibration is callcd a standing wavc, which is widely utilized in dcsign of ultrasonic motors. The modcs posscss thc following important characteristics:
1. Infinite number of natural modes An elastic body is a continuous system with an infinitc numbcr of dcgrces of frcedom. The system possesses an infinite number of natural frequencies (modal frequencics) and modc shapcs, i. c., wc ha vc modal parameters n
(w n , ~n)'
=
1, 2,3, ...
In general, each natural frequency corresponds with one mode shape.
2. Dependence of modal parameters Gcnerally, modal paramctcrs dcpend on mass, stiffness distribution of thc elastic body, and its boundary conditions. 3. Orthogonality of mode shapes Whcn a bar vibratcs longitudinally, from Eq. (4. 5) any of the modcs must satisfy
-.! (E S dx Therefore for modes
(Wi'
~i)
and (w j
d¢) d.L '
p Sw 2 ~
= -
~j)'
~ =
~(x)
(1. 10)
there are
d d.L
(E 'd.L S d¢i)
d d.L
(ES d¢j )S dx - - P
=-
P
S
2-1.
(4. 11)
2-1.
(4. 12)
Wi't'i
Wj 't'j
Chapter 1
Fundamentals of Vibration for Ultrasonic Motors
79
The two end positions of the bar are denoted by 0 and l, respectively. Multiplying Eqs. (1. 11) and (1. 12) by ~j and ~i , respectively, and carrying out integration yields
I' I' o
M)
d ( E5 -d' d.r = ~J -d
o ~i
w;
I' I'
p5~i~Jd.r
(1. 13)
d ( E5 MJ - - WJ2 oP5~J~id.r dx dx ) d.r -
(1.11)
X.T
0
Applying integration by parts to Eqs. (4. 13) and (4. 14), respectively, and using the free boundary condition ( E5M/d.r = 0) or the fixed boundary condition (~ = 0) of the bar, the following results can be obtained:
t
Mi -d MJd x - - Wi'It P5-1.'l'i'l'j -I. d - I E5 -d X o .T.T 0
(4. 15)
t Mi -d Mjdx - - Wj'It P5-1.'l'i'l'j -I. d - I E5 -d X o .T.T 0
(4. 16)
Subtracting Eq. (1. 16) from Eq. (1.15), the remainder is
(w~ -w;)I>5~i~Jd.r
0
=
(1. 17)
Because Wi cFW J ' we have (1. 18a) For a uniform bar we obtain (4. 18b) Comparing Eq. (1. 18a) with Eq. (1.16) gives
MJ dx I to E 5 Mi dx dx
0
=
(4. 19a)
For a uniform bar, there is
MJdx I to Mi d.r d.r
=
0
(4. 19b)
Eq. (1.18) or (1. 19) is the orthogonal condition of mode shapes. More precisely speaking, Eq. (4.18) is the orthogonal condition of displacement mode shapes. Similarly, multiplying both sides of Eq. (1.11) by ~i and then integrating from 0 to l results in
(E5 dd~i )d.r I 'o ~i ddX .T
=-
W;I' p5~; d.r
(1.20)
0
Integrating it by parts and then applying the boundary conditions, there is
Ki Mi
(1.21)
Ultrasonic Motors Technologies and Ap plicalions
80
Ki
I' ,
(dcPi) 2 oES dx d.1':
=
(1. 22) (4. 23)
where Ki and Mi are the ith (order) modal stiffness and modal mass of the bar. respeetively. Sometimes they are also ealled as the ith (order) generalized stiffness and generalized mass of the bar. From Eqs. (1.22) and (1.23). it can be observed that each natural frequency Wi corresponds to both the modal stiffness Ki and modal mass Mi. The three modal properties given above arc universal to the vibration system. not only for bars. shafts. and beams. but also for plates. shells. and more complex vibration systems. 4. Normalization of modes The mode describes the amplitude distribution of an elastic body at corresponding natural frequency. It shows that the amplitudes of all points on the elastic body are not independent. with being proportional to each other. The process used to select the specific ratio or multiples is called normalization. Currently there arc three major methods for normalization L5J : (1) The maximum amplitude of a mode shape is regulated to 1. (2) The modal mass is taken as 1, that is. Mi (3)
J: ¢;
d.1':
=
=
I>S¢;
dx
=
1.
1.
5. Strain modes The stress-strain relation of a bar is (J
=
F
-
S
=
au
E-
a.1':
=
EE
(4. 24)
For the nth (order) mode shape function of the bar. the corresponding strain function of the bar can be deduced as En =
au: a.1'
=
dcPn(X)qn(t) d.1':
=
¢'()
()
n .1': qn t
(1. 25)
¢:
(X) is defined as the nth (order) strain mode shape function of longitudinal vibration or the nth strain mode. 'Table A.2 represents the natural frequencies and corresponding strain mode shape functions of the bar with three boundary conditions. Fig. B. 2 denotes the first four (order) mode shapes of the bar with the boundary conditions. From Eq. (4. 19) the strain modes also possess orthogonality. The modal characteristics mentioned above exist in the natural vibration of bars. as well as in those of shafts. beams. plates. and shells. but which possess different expressions in their displacement and strain mode functions.
4. 1. 3
Torsional Vibration of Shafts
The strut subjected to torsional vibration is often called a shaft. The equation
Chapter 1
Fundamentals of Vibration for Ultrasonic Motors
81
governing the natural vibration of the shaft about its axis can be described by (4. 26)
where e(x, t) = e is the twist angle of the shaft around x axis, Ie (x) = Ie indicates the mass moment of inertia of the unit length of the shaft around :r axis, Ger) = G and J Cd = J represent the shear modulus of the material and the aera moment of inertia of the shaft about :r axis, respectively. For a uniform shaft, Eq. (1.26) reduces to I a'e Oat'
=
GJ a'e
(1.27)
a.x'
When the shaft has a circular section, Io=pJ , Eq. (4.27) becomes
G a'e
a'e at
(4. 28)
p a.x'
Note that Eqs. (4.28) and (4.2) arc mathematically the same. So the characteristics of the torsional and longitudinal vibrations of a bar behave in the same form. Hence detailed discussion of the torsional vibration of the shaft is omitted and only the final results are given in Table A. 3. :'\Iote that the e.xpressions of natural frequencies are only suitable for shafts with a circular section.
4. 1. 4
Bending Vibration of Beam
The strut subjected to bending vibration is often called a beam. We consider the thin beam for which the length is much large than depth (at least 10 times) and the deflections arc small compared to the depth. Then, the rotation of cross sections of the beam is neglected compared to the translation, and the angular distortion due to shear is negligible compared to the bending deformation. Applying Euler-Bernoulli theory, the equation governing the natural vibration of the beam in its lateral direction can be described by
~ (E1 ax'
a'w)+ 5 a 2 w a.x' Pat'
=
0
(1. 29)
where w(.x, t) = w is the lateral vibration displacement of the beam, I (.x) = 1 denotes the area moment of inertia of the beam's cross section about the neutral axis, S(.x), E(.x), and p(.x) represent the cross section area, the elastic modulus of the material, and the mass density of the beam, respectively. For a uniform beam, we have (4. 30)
Letting w(x,t)
=
cp(x)q(t)
(4. 31)
Substituting Eq. (1. 31) into Eq. (1. 30) and using the method of separation of variables, we can obtain the general solution of Eq. (4. 30)
82
Ultrasonic Motors Technologies and Ap plicalions
W(.T, t)
where the constants An and En can be determined from the initial conditions, the constants C and Dn can be evaluated from the boundary conditions, from which we can obtain following characteristic equation, natural frequencies, and corresponding mode shapes for a uniform beam simply supported, respectively: sinX n Wn
=
_ X~ l'
-
0, n
=
(1. 33)
1,2,3'"
{IT '\j ps
(4. 34)
Fnsinpnx
(4. 35)
:;p
where Xn is the solution of the characteristic equation, and pn = Sw~/EI. Natural frequencies and mode shapes of other beams with different boundary conditions can be deduced from some approximate approaches. Table A. 4 lists the values of the first five X~ and characteristic equations of a uniform beam with various boundary conditions. Corresponding mode shape functions are listed in Table A. 5. The first five mode shapes are shown in Fig. B. 3. There are two rigid body modes for the free-free beam, whereas there is one rigid body mode for the free-simply supported beam. The corresponding zero frequencies have not been ineludcd in Table A. 4. In mechanics of materials, the bending moment of a uniform beam and the deflection satisfy the following relationship M(.T,t)
=
E1
a'w(.~,t)
ax
=
E11>"(.T)q(t)
(1. 36)
The strain function corresponded to the nth mode shape function of the beam can be written as E11>': (x)qn (t)z E1
E When z
=
(4. 37)
h/2, Eq. (1.37) becomes En (X)
=
~
f: (X)
(4. 38)
where En (.T) is the nth strain mode of bending vibration of beams. The second derivative of the mode shape functions in Table A. 5 lead to the strain mode of beams, as shown in Fig. B. 1. The strain modes of bending vibration of beams will be widely utilized in design of ultrasonic motors.
4. 1. 5
Natural Vibration of Plates
Various types of ultrasonic motors utilize the natural vibrations of plates- 6- 7J • Therefore, it is necessary to understand the vibration characteristics of plates. Vibrations of plates can be divided into the out-of-plane vibration such as the
Chapter 1
Fundamentals of Vibration for Ultrasonic Motors
83
bending vibration of a plate and the in-plane vibration such as the extention-contraction vibration of a plate. Thc formcr refcrs to the vibration dircction perpendicular to thc platc, whercas the lattcr refcrs to thc vibration dircction within the plane of the plate. This section will discuss these two kinds of vibrations of plates, respectively. According to thc ratio betwecn thickncss h and the smallest sidc length b, , thc plates are classificd as thin or thick plates. Usually whcn h/b 2 ~ 1/20, thc platc can be treated as thin, otherwise, as thick. This section focuses on the natural vibration of uniform thin plates.
1. Vibration equations of uniformthin plates r81 To establish the differential equation of the out-of-plane bending vibration of a uniform thin plate, Kirchhoff assumptions are used in thin plate theory: (1) Because the thickness of the thin plate is small, the out-of-plane normal stress (o-z = 0) can bc ncglcctcd. (2) After bcnding, a linc initially perpcndicular to the middlc plane of thc plate remains straight and perpendicular to the deformed middle plane. This implies that shear deformation can be neglected, i. e. ryz = rxz = 0 and the middle planc is not contracted nor cxtcnded. (3) The rotary incrtia due to bcnding is ncglected. We take the thickness, the density, the elastic modulus, and the Poisson's ratio of the material of the plate as h, P , E, and!1' respectively. Set a coordinate systcm Oxyz in the middlc plane of thc plate, whcre z denotes thc thickncss direction. And thc displacemcnts along the directions of coordinatc are u, v, and w, respectively. Analyzing a small element from the thin plate with the above assumptions, a partial diffcrential equation of thc natural vibration of the platc can bc exprcsscd as i(
h a'w +V(a w +2 ~+ a w)= 0 at' ax' ax' ay' ay' 1
p
1
(1. 39)
where V = Eh 3/12 (1- / ) is the bending stiffness of the plate, and Eq. (1. 39 ) can bc abbreviated as
a'~ + a'
at
V" V'
2W
=
0
(4. 40)
wherc a' = D/ ph, and V" = (a2 /ar' + a' /ay') is callcd the Laplace opcrator. U sing polar coordinates, thc partial differential equation of the natural vibration of a circular plate can be derived as
aw , ( a 1 a 1 ( at' + a ar' + ---; ar + 7 ae' w = 0 2
2
2
)'
(1. 11)
2. Natural vibration of rectangular plates Vibrations of rectangular plates arc discussed here, as some stators or sliders of linear ultrasonic motors utilize such plates[9 10J. Structural parameters of the rectangular plate are length b1 , width b2 , and thickness h. Let the out-of-plane dis-
Ultrasonic Motors Technologies and Ap plicalions
84
placement of the plate as Lll w
=
wCr,y,t)
(1. 12)
~Cr,y)q(t)
=
Substituting Eq. (1. 12) into Eq. (1. 10) and using the method of separation of variables, the following two equations arc acquired
+w
get)
2
q(t)
(4.43)
0
=
a' V" V' ' ~ - w' ~
(1.11)
0
=
If the rectangular plate is simply supported on all of the four sides, the boundary conditions can be written as
f~ 1~
a'~ =0,
=
0,
ax'
=
0,
- , =0,
x
=
0,
y
=
0,
(4. 45)
a'~
ay
y
=
b,
The solution of Eq. (1. 11) can be denoted as
(4. 46)
Csinaxsinf3y
~ =
From the boundary conditions it can be derived that sina b 1 where
am
=
m
7(/ b],
f3n
=
n
=
0,
7(/ b"
sinf3b 2 m, n
=
(1. 17)
0
1, 2,3, ...
=
Substituting Eq. (1.16) into Eq. (1.11) yields the natural frequencies of an uniform rectangular plate simply supported are (1. 18)
and the corresponding mode shapes -I.
(
'l'mn .I,y
)
=
C
. m 7(X. n 7(Y -b- sm -l-
(1. 19)
~mnSln
]
),
Then, we obtain the general solution of Eq. 4. 40: w ( x,y,t )
=
'" '" L..;L..; m-"j
11-]
Cmnsln-b-sm-b. m 7(.I. n 7(Y (A mncoswmnt 1
+ BmnSlnwmnt · )
(4. 50)
2
where the mode shape ~mn (.I,y) consists of m half sine waves in the.I direction and n half sine wave in the y direction of the plate, as shown in Fig. 1. 1 11 '-. We can also use (m -1) and (n -1) for representing the number of nodal lines along the.I direction and y direction, respectively. When m =1 and n = 1, the plate vibrates with base natural frequencyw ll • )Jo nodal line is in the plate, which can be described as ~ll. When m = 2 and n = 1, the plate vibrates with natural frequency W,] , and there is no nodal line in the y direction in the plate but one nodal line in the x direction at b1 /2, which can be described as ~21. Similarly, when m= 1 and n = 2, m= 2 and n= 2, they are corresponding to W 12 and W 22 ' respectively, which can be described as ~1' and ~22' respectively. Figure 4. 1(b) only shows the location of the nodal lines [w(x,y,t) plus/minus signs of the displacements, which are called node patterns.
=
OJ and
Chapter 1
Fundamentals of Vibration for Ultrasonic Motors
85
:EIJ ITIB I
- ----------- - - -----i----- -
-
I I
+
First four modes shapes and node patterns of a uniform rectangular plate simply supported on four sides and dashed lines denote nodal lines.
Fig. 4. 1
In Eq. (1.19) whenb 1 =b" the rectangular plate becomes a square one, whose natural frequencies can be expressed as
Wmn
=
7['
(2
b; m
2)
(D
+ n '\j ph
(4. 51)
From Eq. (4. 51) it is obvious that when m # n, Wmn = Wnm , which implies that these are two identical natural frequeneies (the phenomenon of repeated frequencies) corresponding to the two different (independent) mode shapes CPmn and CPnm. In fact, arbitrary linear combinations of these two mode shapes (CPmn cCPnm) will satisfy Eqs. (4. 44) and (4. 45), which means that these are not two unique mode shapes, but there is a two-dimensional subspace of the mode shapes of the form
+
(1.52) Figure 4. 2 shows a very interesting node patterns obtained from a square plate: 13 ] , where CPl4 and CPo are deduced from Eq. (1.19).
Fig. 4. 2
:"lode patterns of a square plate with two identical natural frequencies
The above example indicates that the mode shape obtained by combining the
two orthogonal mode shapes, which correspond to repeated frequencies W 11 ) ,
(W 14 =
are changed so much with the variation of amplitudes of one of the modes
that they become unrecognizable modes. That is the reason why one should be
Ultrasonic Motors Technologies and Ap plicalions
86
careful and avoid a close pro.7:imity between the interference mode frequencies and the operating mode frequency for ultrasonic motors. For the reetangular plate simply supported on all of four sides, there are analytical solutions. For other rectangular plates with arbitrary boundary conditions only approximate solutions can be obtained. In fact, the rectangular plates frequently uscd for thc design of ultrasonic motors posscss thc frec boundary conditions on all sidcs. Fig. C. 1. shows the first eight displaccment and strain modc shapes of a rectangular plate, which are calculated with A:'\JSYS software. The second column in Fig. C. 1 shows the projections of the mode shapes, also called mode ncphogram. Thc material of thc platc is the phosphor bronze with thc density p = 8.80 X 10-' g/mm', Young's modulus E = 1. 13 X lOll Pa, Poisson's ratio,u = 0.33, and its dimcnsions are that h=2mm, b1 =30mm. and b, = 20mm. From thc node pattcrn it can bc seen that nodal lincs are parallel with the .7: or y axes in general, and all mode shapes are in out-of-plane vibration. For convenience. they are expressed as Bmn , where the subscripts m and n dcnotc the numbcr of nodallincs parallelcd to thc y and x directions, rcspcctively. For thosc nodal lines not paralleled to thc sides of thc rcctangular platc, it is very difficult to express its bending mode using m and n. Therefore graphic expressions are used for the subscripts such as in Figs. C. 1 and C. 2. When bl = b2 = 20mm. the first eight displaccment mode shapcs and strain modc shapes of thc square plate are obtained, as shown in Fig. C. 2. 3. Natural vibration of circular plates[lH5] Many ultrasonic motor stators use circular plates, as shown in Fig. 4. 3. Thcrcfore, it is important to be familiar with the vibration characteristics of a circular plate. Let the radius and thickness of the plate be a and h, respectively. Because of thc symmetry of thc circular platc. thc solution of thc Eq. (4.41) bccomcs w(r.B,t) = R(r) (SmsinmB
+ CmcosmB)c
iw"
m = 0,1,2 ... · (4.53)
Substituting Eq. (1.53) into Eq. (1.11). there are
d2 (dr'
m" + --;1 drd - 7)
1
R(r) - k R(r)
=
(1. 51a)
0
Eq. (1. 51a) can be written as
(.) By PDLab
Fig. 4. 3
(b) By M IT
Circular platc uscd for stators of ultrasonic motors
Chapter 1
[ d2, dr
Fundamentals of Vibration for Ultrasonic Motors
+ ~r ~ _ (k' + m,2) ] [~+ ~ ~ + (k2 - m,') ]R(r) dr r dr r dr r
where k'
=
=
87
0
(1. 51b)
w 2phi D.
The general solution of Eq. (4. 54 b) can be obtained:
where] m(kr) is the first type mth order Bessel function, Y m(kr) is the second type mth order Bessel function, 1m (kr) is the first type mth order corrected Bessel function, Km (kr) is the second type mth order corrected Bessel function, and Am' Em' C m , and Dm arc constants to be determined. For a circular and solid plate, because w andawlar at the center (r be limited value, Em and Dm must be o. From Eq. (4.55) ,we have
= 0) must
Am and C m arc two constants to be determined by boundary conditions. Solving Eq. (1.56) for the circular plate with fixed circumference condition, we can obtain the natural vibration frequency m,n
=
0,1,2,···
and corresponding mode shape function rpmn (r,B)
= =
{] m(kr) - [J m(Amna) I 1m (Amna) JIm (kr)} (SmsinmB Rmn(r)(SmsinmB+CmeosmB)
+ CmeosmB) (4.57)
where m is the number of nodal circles, n is the number of nodal diameters, as shown in Fig. 4. 4_12J; Amn is the natural frequency constant. The first five natural frequency constants for various boundary conditions arc given in Table A. 6[16: when (1=0.3. The above solutions demonstrate that for a thin and circular plate, each natural frequency has two natural mode shapes ()Jote: there is a mode shape only when m = 0) due to structural symmetry of the circular plate. Hence, all the natural modes (except for m= 0), which arc called degenerate ones, breach the general rule that each natural frequeey associates with one mode shape (see Section 4. 1. 2). From Eq. (4.57) the two mode shapes corresponding to Wmn arc given by {
rp~l~ (r, B) =
Rmn (r) Sm sinmB
rp;:;~ (r,B) =
Rmn (r)CmeosmB
(4. 58)
The general solution of Eq(1. 11) can be expressed as w(.:c,B,t)
=
~ ~ [rp~~ (r,B) m-Q
+ rp;:;~ (r,B)Jeiwm"1
(1. 59)
n-O
We can sec that all of the mode shape functions form the two-dimensional linear function subspace, consisting of an infinite number of mode shape functions. In fact, for the two arbitrary and independent mode shape functions, their arbi-
Ultrasonic Motors Technologies and Ap plicalions
88
Nodal line
,
...--::::::
Nodal line
~
Nodal line
,
~~~~ ,
"
First four mode shapes and node patterns of a circular plate with fixed circumference condition, and dashed lines denote nodal lines within the plate
Fig. 4. 4
trary linear eombinations satisfy Eq. (4.41), i. e., the two functions are the base functions in the function subspace. From this we can see that a traveling wave in a special excitation can be yielded in the stator of a disk-type ultrasonic motor because of the utilization of the two independent and orthogonal modes corresponding to one natural frequency, For a thin and circular plate simply supported, the form of the solution similar to that mentioned above can be found by the same method-]3J. However, at present no analytical solutions have been found for thin and circular plates with other boundary conditions. In the design of ultrasonic motors, the thin, circular, and solid plates with a free circumferential condition are commonly adopted. Therefore, in applying ANSYS software, the displacement and strain mode shapes of the first sixteen modes of the plate are calculated and shown in Figs, C. 3 and C.1. The radius of the plate is a=30mm and the thickness is h=1mm, and the material parameters are the same as the rectangular plate mentioned previously. For convenience, those mode shapes containing only nodal diameters are shown in Fig. C. 3 according to the order of ascending frequencies, whereas those mode shapes with both nodal diameters and nodal circles are arranged in Fig. C. 1 in the same order. In Fig. C. 3, the mode shapes with only one nodal diameter represent the rigid body rotation about the nodal line. There is no in-plane strain, resulting in no strain variation in the strain mode shape.
4. Natural vibration of circular ring plates In the actual design of ultrasonic motors, thin and circular ring plates are usually used. For example, Fig. 1. Sea) shows the stator and rotor of a traveling wave type rotary ultrasonic motor, which is used in a system, developed by PDLab. Fig. 4. S (b) shows the stator of TRUM-60 ultrasonic motor manufactured by Trans USM Co. Thus, the designers of ultrasonic motors are interested in vibrations of circular ring plates 117-2o -. A thin and circular ring plate has two concentric circular boundaries, which
Chapter 1
Fundamentals of Vibration for Ultrasonic Motors
89
(b)
(a)
Fig. 4. 5 Circular ring type USM(a) and USM stator(b) usmg a thin and circular ring plate
are at internal diameter b and external diameter a, respectively. There are the total of the nine possible combinations of boundary conditions for thc thin and circular ring plate. Nevertheless, this chapter only deals with the natural vibration of plates with fixed internal circle and free external circle boundaries, due to the fact that most of the stators of disk-type ultrasonic motors are fixed by the internal circle and their external circle are free. In fact, a thin and circular ring plate is the extention of a thin solid plate. We can still apply Eq. (4.41) to describe the natural vibration of the ring plates. Substituting Eq. (4.53) into Eq. (4.41), considering the above boundary conditions of the plate and solving Eq. (1. 51 b), we can obtain Wmn
=
~~n
Iff
(1.60)
where Amn is called the natural frequency constant of the out-of-plane of the thin and circular ring plate. Table A. 7116 - lists the first fifteen natural frequency constants Amn of the out-ofplane of a thin and circular ring plate with the fixed internal and free external circumferences when (b/ a) = O. 1, O. 2, and O. 3. It is difficult to find an analytical solution of the modes of the plate; only a numerical analysis can be used. Figs. C. 5 and C. 6 show the first sixteen displacement and strain mode shapes for out-of-plane vibrations of the plate with the fixed internal and free external circumferences boundaries using A='JSYS software. The inner diameter of the plate b =34mm, outer diameter a=60mm (b/a=O. 57), and its thickness h=4. 6mm; its material is phosphor bronze, whose characteristics have been given in the retangular and thin plate calculated above. It is similar to the thin, circular and solid plate mentioned above: its mode shapes containing only nodal diameters are arranged in Fig. C. 5 according to the order of ascending frequencies, whereas its mode shapes with both nodal diameters and nodal circles are arranged in Fig. C. 6 in the same order. These charts are extremely useful for the stator design of disk-type USMs.
5. In-plane natural vibration of thin plates The out-of-plane natural vibration of uniform thin plates
IS
discussed above, m
90
Ultrasonic Motors Technologies and Ap plicalions
which the displacement wer, y, t) in the z direction is considered. and in the .1': and y directions the displacements arc u(x. y, t) = v(x. y. t) = o. The so called in-plane vibration of a thin plate means that its displacement w(x, y, t) = 0 in the z direction, but the displacements in both the :r and y directions exist. In reality. the natural vibration of a thin plate can generate displacements in the .1':, y and z directions. However at low frequencies. its vibration direction is mainly out-of-plane, and the in-plane displacements can be ignored; at high frequencies, the in-plane displacements in the .1': and y directions play primary role, and the out-of-plane displacement (in the z direction) can be neglected. The in-plane vibration of a thin plate possesses the multipe types: extensioneonctraction, bending and torsion, which always exist at the high-frequency range no matter whether the plate is rectangular or circular or annular. Although the design of ultrasonic motors mostly focuses on utilizing out-of-plane vibrations of thin plates: 21 22J , as shown in Fig. 1. 6 (a), making use of in-plane vibrations of thin plates plays an increasing role currentlyL23-25-. The rotary ultrasonic motors utilize in-plane bending vibrations of circular plates or circular ring plates. as shown in Fig. 4. 6(b), whereas linear ultrasonic motors utilize the extension-contraction vibrations and bending vibrations of rectangular plates[2627:, as illustrated in Fig. 1. 7. Recent research indicates that the ultrasonic motor will be more compact utilizing the in-plane vibration mode of thin plates.
(a) Our-o r-plane bending mode shape B 07 of tator
Fig. 4. 6
(b) In- plane bend ing mode shape B" of stator
Stator of traveling wave type rotary ultrasonic motor using thin ring plate
The in-plane vibrations of thin plates arc more complicated than out-of-plane ones. and we only discussed the in-plane vibration of a thin and circular plate here. Ref. [28J, as the expansion of elastic bar vibrations, describes the in-plane extension-contraction vibrations, torsional vibration, and out-of-plane bending vibrations of the circular ring consisting of a bar. In Ref. [28J, the cross-sectional dimension of the bar is much smaller than its average diameter of the circular ring, although the shape of its cross-section is not limited. However. in the design of ultrasonic motors, the thin and circular plate used for stator has a round hole in its center. In many cases, its cross-sectional dimension and average diameter arc of the same order of magnitude. In-plane vibration problems of the uniform, thin. and circular ring plate have been studied in some monographs and papers; readers can refer to Refs. [28-29J
Chapter 1
91
Fundamentals of Vibration for Ultrasonic Motors
y(v)
b
2
x (u)
,}
z (ft)
hI
(a) TIlln rectangle plate used for linear ultrasonic motor
/0;
z
(b) First longitudinal mode shape of stator in xOy plane
Fig. 4. 7
(c) Second bending mode shape of stator in xOy plane
Stator of linear ultrasonic motor of using thin rectangle plate
The stress and strain analysis of in-plane vibrations of thin plates belongs to plane stress problem in mechanics of elasticity. In rectangular coordinates, the circular plate displacements are that U = U(.T, y, t), v = V(.T, y, t), w = 0, and thc strcsscs (J3 = T32 = T31 = O. Using the strcss-strain rclationshipcoo:, and assuming that intcrnal and extcrnal circumferences of a thin and circular ring platc are both free, we can derive the natural frequency of the in-plane vibration of the plate: (1. 61)
where m represents the number of nodal circle, and n represents the number of wave or nodal diameter. Table A. 8- 11J lists the value of Amn which is the natural frequency constant of a thin and circular ring platc with the above boundary conditions: 3 !], taking E= 7. 871X10 11 Pa, p=7.73g/cm 3 , andf1=O.32. The above analytical solutions requirc that thc plate must bc homogcneous and isotropic. When the stator is made by combining different materials, it is not appropriatc to apply simply analytical solutions mcntioncd above for quantitativc analysis. It is vital to dctermine the cquivalcnt homogcneous material paramcters, which may be rather difficult in practice. For a homogeneous and isotropic thin and circular plate, the modal analysis is carried out for the in-plane vibration of a thin and circular ring plate with freefree boundary conditions, obtaining first 16 modc shapes arc listcd in Figs. C. 7 and C. 8, from which and the in-planc modc shapes givcn in Ref. [32J it can bc secn that in the natural vibration of thc homogcncous, isotropic, thin and circular ring plate, in gencral case the displacemcnt of every point possesses both radial and circumfercntial components. In thc design of ultrasonic motors, thc ra-
92
Ultrasonic Motors Technologies and Ap plicalions
dial component can be used to provide the condition for applying pre-pressure, which can lead to reduction in sizes of an ultrasonic motor. Fig. 4. 8 shows an ultrasonic motor utilizing the in-plane bending vibration of the circular ring stator which is designed by Chunsheng Zhao and H ui GuO[20:. Figure C. 9 shows the first eight displacement and strain modes shapes of inplane vibrations of a thin and circular ring plate with the fixed internal and the free external boundary conditions, which arc obtained with ANSYS software, among which Boo, BOl , Bo2 , Bo3 , and B" are in-plane bending mode shapes, and E 01a ' E oo ' and E 02a are in-plane extension-contraction mode shapes. Fig. C. 10 shows the first eight displacement and strain modes of the in-plane vibration of a thin and circular solid (b=O) plate with a free external circumference condition.
(b) Ullraso nic motor usin g mode shape shown in Fig.4.8(a)
Fig. 4. 8
Ultrasonic motor based on in-plane vibration
There is no certain rule for the value of frequencies and their odcr of appearance. In the in-plane natural vibration type (extension-contraction, bending and torsional vibrations) mainly depends on the structure of the thin plate and its material parameters. For the same thin and circular ring plate, its out-of-planc vibrations often occur in the low frequency band, and its in-plane vibrations occur in the high frequency band. Figs. C. 3, C. 1, and C. 10 show the out-of-plane and in-plane mode shapes for the same thin, circular and solid plate, respectively Figs. C. 5 and C. 6 denote the out-of-planc mode shapes for the thin and circular ring plate. Figs. C. 7, C. 8, and C. 9 denote the in-plate mode shapes for the same thin and circular ring plate. These figures illustrate that there is no certain rule in mode shapes and vibration types.
4. 1. 6
Natural Vibration of Cylindrical Shells
Some stators of ultrasonic motors are designed with cylindrical shells (cylinders) L33·36.. Their wall thickness is smaller compared with their radiuses; therefore, they belong to the category of shells. The cylindrical stator's vibration can be analyzed using the shell vibration theory. The vibration of a shell is complex because it has three displacements for every point. )Jo universally recognized or
Chapter 1
93
Fundamentals of Vibration for Ultrasonic Motors
unified theory has been found. In a variety of shell vibration theories, for the elassieal thin-shell vibration theory with small displacement, Love assumption has been widely adopted:"].
1. Vibration equations of cylindrical shells A shell is made up by two adjacent curved surfaces; the surface of the same distance apart from these two surfaces is called neutral surface, and the shell's thickness is the vertical distance between the two surfaces. Love assumptions: (1) The thickness of a shell is smaller than other dimensions (for example, the smallest radius of curvature of the neutral surface). (2) Both strain and displacement are sufficiently small, so that the second and higher order variables can be neglected in strain-displacement relations. (3) :'\Jormal stress along the z direction in comparison with that of other directions can be neglected. (4) A straight line normal to the neutral surface before the deformation remains the straight and perpendicular to the neutral surface after the deformation. Let R, h, and l denote the radius of the neutral surface, the wall thickness and the length of the cylindrical shell, respectively. .1': and are selected as the shell surface coordinates, as shown in Fig. 4. 9. For convenience the dimensionless coordinate along the length direction is adopted:
e
x/R
5 =
Fig. 4. 9
(4. 62)
Cylindrical shell's coordinate system and geometrical parameters
Let
u
[u
=
v
wy
(4. 63)
where
u
e, t) e, t) w(s, e, t)
=
u(s,
v =
v(s,
w
=
Eq. (1.63) represents the displacement column matrix of any point on the shell surface. The natural vibration equation of the cylindrical shell becomes lJIu
=
{a}
(4. 64)
Ultrasonic Motors Technologies and Ap plicalions
94
where lJI is the matrix operator. For vanous shell theories. lJI possesses different forms. For example. the operator from Donnell-Mushtari theory can be expressed as
~
[as 2 + _
(1-
p
lJID -
M
~~ Z
aB'
1+ _a'_ !:.2J!:.
/)R 2 a' Eat' ]
asaB
Z
a
liEL z asaB
=
aB
[1 + k V'
a
aB
+p
(1- /)R'
a2
Eat'
]
(1. 65) where \7' =\7 2\7 2. \7' =
a2las' + a2laB 2 • k = h'/IZR 2 •
2. Natural vibration of cylindrical shells with finite length [8J For the cylindrical shell with finite length and two free ends (the cylindrical shell stator is usually in this case). the displacement functions can be assumed as (s) cosnBcoswt JI uv AX: EX (s) smnBcoswt =
=
lw
=
(4. 66)
k
ex
k
(s) cosnBcoswt
where X k (s) is the displacement function of a beam. whose two ends possess the same boundary conditions as the cylindrical shell. It is also the kth mode shape function in thenatural vibration of the beam. X~ (s) is the first derivative of X k (s) with respect to s. Substituting Eq. (4. 66) into Eq. (4. 64). we can obtain A. Band C and the natural frequencies of the shell. The mode shapes obtained in this way are usually coupled in three directions. :'\Jodes locate along circumferential direction and nodal sections perpendicular to axis exist along the axial direction. as shown in Figs. 4. lO(a) and (b). respectively. In the figure. m represents the number of the nodal sections. and n represents the number of circumferential waves. Fig. 4. 11 shows the first six mode shapes which possess relative large radial displacement in the cylinder. The results are obtained with A:'\JSYS software. and the boundary conditions of the cylinder are free at both ends. The structural parameters are listed in Table 1. 1. Table 4. 1
Structural parameters of cylinder
b/mm
a/mm
l/mm
p/(g/ em')
26
31
30
2. 7
In Fig. 1. 11. the mode shape B 02 has no nodal cirele along axial direction. The
Fundamentals 01 Vibration Ior Ultrasonic Motors
Chapter 1
" $" CD , --~
\
, ....... __ ... ,
0-\\
95
~\
1,'
~--
Ca) Nodes along circumferential direction
I~--------------~I -------------------~---~ -~:=::~ -....... ----.. ...... _-Cb) Nodal sections alond axial direction
Fig. 4. 10
B.x.
O.8016kHz
BIlJ 22.304 kHz
Fig. 4. 11
Displacement mode shapes of a cylindrical shell
B"
IO.154kHz
B" 25.538kHz
B"
41.25 1kHz
BlJ 44 .251 kHz
First six displacement mode shapes 01 a cylindrical shell stator
mode shape B03 is the same as the B 02 ' This feature will be used for non-contact type ultrasonic motors in Chap. 11.
4.2
Forced Vibration of Elastic Body
The vibration of a system or structure under external excitation force is defined
Ultrasonic Motors Technologies and Ap plicalions
96
as a forced vibration, and corresponding response is defined as its forced vibration response or dynamic response. The stator (or slider) of ultrasonic motors is excited by piezoelectric ceramic elements. In this section we mainly study the forced vibration response to the piezoceramic excitation. Both longitudinal-torsional hybrid ultrasonic motors and linear ultrasonic motors utilize longitudinal forced vibration responsesL37-39_. Therefore, it is necessary to study the longitudinal forced vibration of bars. The natural vibrations of the elastic bodies without dampings have been discussed above. However dampings always exist in practical elastics bodies, thereby forced damped vibrations of bars arc discussed in this section.
4. 2. 1
Response of PZT Bar to Distributed Electric Field
The bar-type ultrasonic motor, whose stator is similar to a bar, utilizes PZT elements(pieces) to excite its vibration. The excitation by PZT elements belongs to the strain excitation category. It relics on the inverse piezoelectric effect of PZT ceramic pieces producing strain, which excites the stator. First, we study a particular case: the longitudinal vibration response of a PZT bar to the distributed electric field. Assuming that the PZT bar is a uniform and polarised one along x direction. A distributed electric field is applied to the bar, as shown in Fig. 4. 12, and the electric field intensity can be expressed as
E3 (x, t)
=
Eo (x) eiw '
(4. 67)
E,(x,t)
----i x(3)
Fig. 4. 12
A PZT bar excited by distributed electric field
If the displacement of the bar is u(x, t) , then u(x,t) =
~CPi(X)qi(t)
(4. 68)
i=)
The distributed strain of the bar can be written as E3
(
x,t
)
=
3u(x,t)
3.-r
(4. 69)
Using the second piezoelectric equations (sec Table 2. 2), we have {
a 3
C-r,t)
==--
D 3 (x,t) -
e33E3
C-r,t) ~
e33 E 3 (x,t)
The potential energy of the bar is
Cf3E3
+(33E3
C-r,t)
(x,t)
(1.70)
Chapter 1
Fundamentals of Vibration for Ultrasonic Motors
I'
97
I' ,
1 oSE,(.:r:,t)D 3 (.:r:,t)d.:r: V= Z1 oS(J,(.:r:,t)c,d·:r:- Z
(1.71)
Substituting Eqs. (4.68), (4.69) and (4.70) into Eq. (4.71) yields
V=
where k i}
=
J:
-e 33 SE 3 (.:r:,t)t¢:(.:r:)qi(t)d.:r:+
•
t¢~(x)qj(t)dx- ~I>~3SE;dx
J: -
qi(t)
-
~I>;3SE32(.:r:,t)d.:r: k}i
=
lectric bar under
J:
ei3 S
E=
¢: (x)dx+
e33 E 3(x,t)S
t
~I>f3S[t¢:(.:r:)qi(t)J ~
t
~ k i) qi(t) q} (t) (1.72)
¢: (x) ¢; (x) dx,
(~3
is the dielectric constant of the piezoe-
constant.
The kinetic energy of the bar excited by the distributed PZT pieces can be written as } '=
~I' S[au(x,t)]'d 2 oP'
at
(1. 73)
.:r:
Substituting Eqs. (4.68) and (4.69) into Eq. (4.73) produces T
=
~ I>S [ t ¢i (.:r:)
ili (t) J
=
~ ~ ~mi} qi(t)
il}(t)
i=l
where m ij
=
m ji
[~¢} (.:r:) il} (t) Jd.:r: (1. 71)
j=1
I>S ¢i (x) ¢j (x) dx.
=
Considering damping C, the dissipated energy is introduced:
D where C i}
=
e}i
=
J:
=
~ ~ i=ci} qi(t) il}(t) i=l
(4. 75)
;'=1
C¢i (x) ¢j (x) dx.
Discussing only dynamic response of the bar, we can neglect the effect of charges on the electrodes, substitute Eqs. (4.72), (4.74), and (4.75) into the following Lagrange equation (4. 76)
The longitudinal vibration equation of the bar with distributed electric field can be obtained:
Mij(t) +Cq(t) +Kq(t) where M
=
[m i}] , C
=
[e i}] , and K
=
=
F(t)
(4. 77)
[k i}] are the generalized mass, damping,
and stiffness matrices of the bar, respectively. F(t)
=
[J:
Se33 E3 (.:r:, t)
¢/ (.:r:) d.:r:]
Ultrasonic Motors Technologies and Ap plicalions
98
is the column matrix of the generalized force. Utilizing the orthogonality of displacement and strain mode shapes, i. e., Eqs. (1.18) and (1.19), we can obtain (1.78) or
(1. 79) where M n , C n, and Kn are the nth modal mass, modal damping, and modal stiffness, respectively. The modal force is
Fn(t) where Fn
=
=
2f: S e 33 E 3'«(x)dx 2 f: Se33 Eo (.z:)
2e iW 'f: Se33Eo(xH'n(x)dx
=
=
Fne iw ' (4.80)
¢: (.z:) d.z: is the amplitude of the nth modal force,
¢: (x) is the strain modal function of the bar.
and
The steady-state forced response of the PZT bar to the distributed eleetrie field is
u(x,t)
=
2:;Fn¢n(x)eiCwt-~) I[K n ~(l_W~)2
+ (2SnwY]
"-I
(4. 81) n=l
where
An
=
Fn¢n (.z:) 1Kn
En
=
[~(l_W~)2
~n =
Cj 2Mu K n
tang'n
=-
+
(2Sn wY
Tl (1. 82)
2 ~n wj (1 - w~)
The following conclusions can be learnt from Eq. (1.82): (1) Under the excitation of a generally distributed electric field, the steadystate foreed response of the bar is the superposition of all modal responses. (2) When Wn = 1, this bar enters the nth longitudinal resonance. At this time, the steady-state foreed response of the bar is mainly the nth modal response.
(3) When the distribution of Eo (x) is proportional to the stain mode, that is (1. 83)
Then the modal force can be written as
Fn
=
2e33 KnEo 1E
(1. 81)
where E is the clastic modulus of material of the bas. :'\Jow the nth "pure" modal response of the bar can be obtained
Un (.z:,t)
=
AnBneiCwt-n/2l
=
2e"¢n (.z:)EoIE[ ~(l- W~)2
+
(2Sn Wn)2 ]eiCwt-n/2l (4. 85)
Chapter 1
Fundamentals of Vibration for Ultrasonic Motors
Displacement mode shape
Strain mode shape
99
Electric field intensity E,Cx)
x(3)
Fig. 4. 13 Distribution curves of first displacement and strain mode shapes of a PZT bar and that of electric field in tensity along longitudinal direction
It can be seen from Fig. 1. 13 that in general, excitation forces (force excitation method) must be distributed in accordance with the mode shape in order to excite the first longitudinal" pure" mode shape, which is shown by the fine dotted line in Fig. 1.13. If the distributed electric field (strain excitation method) is utilized to excite the first longitudinal "pure" mode shape, the electric field intensity E, (x, t) applicd to thc bar must havc thc distribution in accordancc with thc strain modc shapc, as shown by thc bold dottcd lincs in Fig. 4.13. Evidcntly, thcsc two cxcitation mcthods arc vcry diffcrcnt. This is the reason why the author especially emphasizes the strain mode shape, and lists strain modes of bars, beams, and plates in the above section, and part of them are often employed in the design of ultrasonic motors. (1) In fact, it is impossible to apply a distributed electric field to the PZT bar along :r axis according to the strain mode shape. In general, the PZT pieces are only placed at a few points (or a few short segments) along the longitudinal direction of a metallic bar and employed as excitation sources.
4.2.2
Metallic Bar Excited by Single or Multiple PZT Pieces
Thc Langcvin vibrator is thc typical cxamplc of cmploying singlc or two PZT pieces to excite a bar. As shown in Fig. 1. 11, two (or few pieces in parallel) PZT pieces are clamped with two metallic cylinders at .T o of the bar. Neglecting the influences of joints bctwccn thc PZT picccs and bctwccn thc two cylindcrs on elasticity of thc bar, and assuming pp Sp = pm Sm = pS, then we can consider the bar as a continuous and uniform one.
Ot----t x(3)
Fig. 4. 14
Sketch of a metallic bar excited by two PZT pieces
Ultrasonic Motors Technologies and Ap plicalions
100
Let us impose an electric field intensity to the each PZT piece (4. 86) where Eo = constant for the thin thickness of the PZT piece. Applying the unit pulse function B(.T - .T o ) with the characteristics B(.T -
.T o )
{
0, 00 ,
x
Xo
=
then from Eq(1. 80), for free-free boundary conditions we can obtain the modal force
Fn (t)
=
2f t0 S e 33 E 3 ( X, t ) B( x -
Xu
)-1.' ( )d X = 'I' n X
33 T[E 2 -Sne . nT[X u i - [ - - a sIn - [ - e
w'
(4. 87)
h were Fn
=
- 2Sne" T[E [
. n T[Xo
oSlil - [ - .
Thus the steady-state longitudinal vibration response of the bar to the PZT pIeces IS U(.T, t)
=
~
-
2S[ne" T[Eo sin
n~To cos n;.T ei(w'-~n) /
[Kn
J(l - iLD' + (2 Sn iLl n)' ]
n-1
(1. 88)
where
(1.89)
15t longitudina l slrain shape
2nd torsional stmin shape
Isl longitudinal \TIlin hape
2nd torsional strain shape
~ i .:--.. ..;:l-------! /:
! \'"
. .~.;;:~...-_.. --~ -----~·-i:-- _.....~-...---. ~:. ..
,' : :
......
.. ..
"
,
•
I
I I
I I
L~__ .•_____ ._.~::~~!!.!":_':~_._L ---~ --J-------_____. Longitudinial Torsional excitation PZT excilation I'ZT
I
I
Fig. 4. 15 Typical example of PZT pieces' installation positions in Langevin vibrator
Chapter 1
Fundamentals of Vibration for Ultrasonic Motors
101
The following conclusions should be emphasized: (1) (sin n~xO)mcx -
(An)mcx' Bccausc of the first longitudinal strain
(Fn)mcx -
mode shape of the bar with free-free boundary conditions. [sin(n IT.L o / l) Jmcx occurs at Xo = l/2. Thcreforc. in ordcr to obtain bcttcr excitation cffect in thc longitudinal vibration (also in the torsional vibration). the PZT picces must bc placed in the middle of the Langevin vibrator[
=
l . n IT.L o -2 • sln-l-
=
{±
O.
1, n = 1, 3 • 5 • ... . . _ 2 ... whIch leads to the ampbn .4.6.
tudc of modal force being F, = F, = F6 = ... = O. Thus. thc 2nd. 4th. 6th longitudinal modes. etc. cannot be excited no matter how much electric field intensity is applicd to PZT pieccs. (3) Under normal condition. the single or two PZT pieces can excite low order "pure" modes in the longitudinal vibration of the bar. as long as its location is appropriately chosen. Whcn the lcngth over diametcr ratio of the bar is greater than 3. longitudinal modc frcqucncics kecp a largcr distance from cach other and do not interfere each other.
4. 2. 3
Response of Beam to Constant Electric Field Intensity
A uniform metallic beam is sticked by a thin PZT strip. on which imposing a voltage Vet) = Vo e iwt • and the PZT strip is uniform along its axis direction. as shown in Fig. 4. 16. Since the thickncss hI' of the PZT strip is vcry thin. thc avcragc electric field intensity is (4. 90) wherc Eo
=
Va / hI'
=
constant. z(3)
y(2)
h
Fig. 4. 16
A metallic beam excited by a thin PZT strip
Assuming that displacement of the beam in bending vibration can be expressed as
2.: ¢,(x) q,(t)
w(x.t) =
(4. 91)
At the same timc. the PZT strip is only affixed on thc top surfaces of thc beam. Therefore. the strain of the PZT strip is h 3'w(x.t)
£1 """
2
3.L'
(1.92)
102
Ultrasonic Motors Technologies and Ap plicalions
»
where h (including hp ) is the beam's height, and h hp; pp ~ pm = p; pp and pm denote the mass density of the PZT strip and the metallic beam, respectivcly. According to the second piezoclectric equation (see Table 2.2), it can be obtained that
J Cr, t) lD, (x,t) 0"1
==
e'l E,
e'lE1
+
Cf1 E1
(4. 93)
+ ,;,E,
the strain energy of the distributed PZT strip can be written as V p --
bhpf'00"1 ( .7:,t ) E1 d.7:- 2 bhpf'0 D ,.7:,t ( )E,.7: d -2-
(1. 91)
where b is the width of the PZT strip and beam. Substituting Eqs. (4.91)-(4.93) into (4.94) yiclds the strain energy of the PZT strip V p = b h1 h,t - e'l E,
.[t sb'~
t sb':
+ bh8h't
(.7:)qi (t) d.7:
b~h,t e
(x)q; (t) Jdx -
31
E3
t
Cf1
[t sb':
(.7:)qi (t)
¢i" (x) q,(t) dx -
J
b~'J~3 E; dx
2d bhhp~ ()f' bh pf"E -2-ftqi t 0 -e3l E%'()d 3'1'i·7: .7: +l~kPi;qi () t q; () t --20(333.7:
2ft
(1. 95)
wherek P l}
k"
=
}l
=
p ft0 bhh 2 ellE sb"( x )sb"( ) x )dx. 1
In addition, the strain energy of the beam (excluding PZT strip)
=
IS
~ ~ ~k~ q,(t)q/t) z-j
wherek~
k7i
=
(1. 96)
)-1
=
J:EI¢';(x)sb';(x)dx.
The total strain energy of the beam in bending vibration is therefore (1. 97)
The kinetic energy of the beam (including PZT strip) in bending vibration can be written as T
=
=
~ f>S [c: (x,t) ~
i= i-j
wherem ij =m;i
=
r tt dx
=
i=miJli(t)qj(t)
~
qi(t)q; (t) f>S¢i(xH; (x)dx (4.98)
;'-1
f>S¢i(xH;(x)dx.
Taking the actual damping into account, the consumption function can be expressed as
Chapter 1
Fundamentals of Vibration for Ultrasonic Motors
D where C i}
=
C}i
=
J:
103
~ ~ ~ Ci}ej,(t)q} (t)
=
I-J
(4. 99)
)-1
C¢i (x) ¢j (x) dx.
Substituting Eqs. (4.95) - (4.99) into the Lagrange Eq. (4.76) produces Mij(t) +Cq(t) +Kq(t)
=
(1.100)
F(t)
where M = [m i} J, C= [e i} J, and K = [k i} ] are the generalized mass, damping and stiffness matrices of the beam, respectively. kij = k;; kt, F(t) = b hh"
. [J:
+
e3l E3 rfl: (.T) d.TJ is the column matrix of the generalized force.
According to the orthogonality conditions of displacement and bending strain modal shapes of the beam: 1 '] , we can obtain
or (4.101) where M n , C , and Kn are the nth modal mass, damping, and stiffness of the beam, respectively, and its modal force is
Fn (t) = bhh p
f:
e31 E3 rfl}n (.T) d.T = bhe 31 e iwt
where
Fn
=
bhe 3 1
J:
J:
Vo rfl: (.T) d.T = Fn e iwt
Vo rfl~ (x) dx
(1.102)
(4.103)
Subjected to the excitation of the PZT strip, the steady-state bending vibration response of the beam can be expressed as w(x,t) =
2.:Fn¢n(X)/[Kn
Jo-w:) + (2Sn wj
]eiCwt-.n)
11=]
(4.104) 11-]
where (4.105) Comparison of Eqs. (1.105) and (1.82) reveals that they are identical in formation, and thus the same eonelusions can be drawn, which are not to be repeated here. It is important to note the followings: (1) The content of the modal force amplitude Fn of Eq. (1.102) is different from that of Eq. (1.80).
If Vo
=
EN'~ (x) =
Vex) n
=
m
n
=F
m
(1.106)
104
Ultrasonic Motors Technologies and Ap plicalions
Then, the mth pure modal response of the beam can be excited, and the response is proportional to the width of the PZT strip b, the eonstant e31 , and the height of beam h. (2) It is known from Eq. (1. 106) that in order to obtain the "pure" mode of a uniform beam, distribution of V(.L) must be the same as the strain distribution. It ean be learnt by comparing Figs. B. 3 with B. 4 that among four typical boundary eonditions, the displacement mode and strain modes are identical only in the simple supported beam, whereas in the other three conditions the two kind of modes are different. (3) In reality, it is difficult to impose a distributed voltage identically to strain modal function of a beam. In genera), a number of PZT strips (pieces) are affixed on a beam to approximately achieve "pure" mode excitation.
4. 2. 4
Excitation of Simply Supported Beam by PZT Pieces
As shown in Fig. 4. 17, the simply supported beam is excited by one PZT strip (piece). Let a,b, and hI' denote the length, width, and thickness of the PZT piece, respectively, and the characteristic coordinate of the excitation PZT is .L o • The excitation effect of the PZT piece at point .Lo is expressed as "distributed foree" by the unit pulse function with variable (x- x o ). For a simply supported beam, we can obtain from Eq. (1. 102): Fn(t)
=
bhe3IVoeiw'J:B(x-xo)1«x)dx 2
=-
nIT) . nIT iw' b h eol V 0 ( T sIn Txoe
(4.107)
xo
h
0
xCI)
g"""
,d; ~ _.
;1 ~
~E3 I
Fig. 4. 17
A metallic beam excited by single PZT piece
Thus, the steady-state bending vibration response of the beam to one PZT piece excitation is 2
ex:
W(.L, t)
=
~
-
b h e 31 Vo (nt) sin nlIT.Lo ( sin nlITx)
n=l
I[K n ~(l-w~) =
~ AnB nei(w'-.n)
+
(2l;"n wj
Jei(wt-~n) (4.108)
Chapter 1
Fundamentals 01 Vibration Ior Ultrasonic Motors
105
where
{
An
=-
En -
bh
e VO (T) 3l
(,jCl - iLI~)
2
sin
TXo sin nZ'I[x
+ (2Sn iLlY)
(1.109)
The form of Eq. (4.89) is similar to that of Eq. (4.109) about the longitudinal vibration response to one single point excitation, and hence related conclusions are not repeated here. The displacement and strain modes possess the same shape in the simply supported beam. Therefore, one PZT piece is placed wi thin the range of the half wavelength of its mode, the polarization direction of the PZT matches with the "+" and" - " of the mode, and PZT piece's width beam's width and its length a ,1./2. as shown in Fig. 1. 18.
b<
<
Extension
Contraction
c::!::!::!:I
Pic7..oeiecfric ceramic piece
+ , - Vibration direction of beam Contraction
Extension
Fig. 4. 18 Sketch 01 a simple supported beam excited by PZT multi pieces
'if".(],.
Po larization direction of PZT
Fig. 4. 19 Distribution 01 PZT pieces exciting a simply supported beam
For the simply supported beam. exciting the first mode only needs one PZT piece; for the second mode, two PZT pieces are required, and for the third mode, three PZT pieces are needed. and so for the forth, as shown in Fig. 1. 19. It can be learnt from Eq. (1.107) that when PZT pieces are placed at the locations corresponding to the largest amplitude of the strain mode shape, the generalized force achieves the largest value, and the biggest strain energy is transferred to the beam, leading to the best excitation effect. If the PZT piece is affixed to the entire range of the half wavelength, the part of PZT piece elose to the wave peak has the best excitation effect, whereas that apart from the wave peak has a less one. When the PZT piece is just located in the nodal point (or nodal line) of the strain mode, the excitation effect is null. Thus, according to this analysis, it is unnecessary that the length of the PZT piece a is just equal to A!2. but may be < ,1./2, which can still achieve good excitation effect.
106
4. 2. 5
Ultrasonic Motors Technologies and Ap plicalions
Response of Thin Plate to PZT Piece Excitation
It is known from section 1. 1. 5 that the natural vibration of a thin plate can be divided into two categories: in-plane and out-of-plane vibrations. Each category can be further divided into three vibration types: bending, torsional. and extension-contraction vibrations. Every type has an infinite number of natural modes. The vibration response of a uniform plate under the general excitation force is the superposition of the all modes of three in-plane types as well as three out-plane ones. Therefore, in order to obtain the required" pure" mode, PZT pieces must be properly configured. It can be learnt from the above analysis that PZT pieces can only be affixed to the upper or lower surfaces of the thin plate. and can excites the plate's vibration based on the reverse piezoclcctric effect. The correct layout of PZT pieces on the thin plate is a precondition in order to effectively excite required modes. The results in previous sections can be applied to uniform thin plates. For inplane or out-of-plane vibration of a rectangular or circular plate. the PZT pieces must be distributed in accordance with their strain modes. In the appendixes of this book, a number of out-of-plane and in-plane mode shapes for thin plates have been provided. Fig. 4. 20 shows several typical examples about the layout of PZT pieces. Nephograms of Stcan mode shapes
ZT pieces Layouts ( posit ion) o f P
iJ" <J,.
Polarization direction of PZT pie "es
:-.-::: .: ~ 0!0; . . ::--
1-
~
.••..•..;d•.. . . .•.
Fig. 4. 20
Typical layouts of PZT pieces used for thin plate vibration
The following conclusions should be noted especially: (1) It is known from the section 4. 1 that for a thin rectangular plate simply
Chapter 1
Fundamentals of Vibration for Ultrasonic Motors
107
supported in all four sides. its displacement and strain mode shapes of the out-ofplane vibration are similar. The two types of modes of the out-of-plane vibration of a thin and circular ring plate. with the fixed internal circumference and free external circumfercncc conditions. arc also similar (sec Figs. C5 and C6). (2) Bccausc thc strain mode shapes in the ultrasonic frcquency band can not bc measured at prcsent. piezoelcctric ccramic pieccs can bc distributed with thc principle summarized above. i. e. the layout of PZT pieces on the thin plate can be designed based on displacement and strain modes provided with this chapter. (3) Both the extension-contraction and bending vibrations of a thin plate can bc cxcitcd by d 3l or e 3l of PZT picces. To thc cnd. PZT pieces nccd to bc distributcd with a specific structurc. Thcy arc placcd at thc appropriatc locations through employing the above conelusions.
4.3
Wave Propagation in Elastic Body
Ultrasonic motors arc dividcd into two catcgories: a standing and traveling wavc typcs. The former dircctly utilizes the modes (standing wa vc) of an elastic body. whercas the lattcr utilizes thc rotation modc (traveling wave) synthesized by two particular standing wavcs. Thc standing wave belongs to the subjcct of mechanical vibration. and thc traveling wavc belongs to the subject of elastic wavc. Wave thcory is also onc of elassic scicnccs. and therc arc many works[12,15,,,,,:. This section focuses on the wave theory used for ultrasonic motor design and only introduces basic concepts and a simple wave theory. without involving acoustic. light. and electromagnetic waves.
4.3.1
Basic Concept of Wave
A substance transmitting wave is known as medium. The change of the motion of the substance with time is called a vibration. The change of the motion of the substance with both time and space is called a wave. Any localized disturbance in a medium can be transmitted to other parts of the medium through the phenomenon of wave propagation. Thc sprcading of ripplc in a watcr pond. the transmission of sound in air and thc propagation scismic trcmors in Earth arc examples of wavcs in diffcrcnt media. Thc transmission of a sine wave along the direction of the axis .1': and with respect to time can be regarded as the simplest example of wave motion. which can be expressed as the function of distance.1': and time t. u(x.t)
=
Usin(kx - wt)
(4.110)
where U is the amplitude. and k is the wave number. In addition. letf'. T. A. and c denote the frequency. period. wavelength. and wave propagation velocity (phase velocity) in the medium. respectively. The socalled ultrasonic wave is its frequency more than the upper limit of the audible sound frequency. This upper limit frequency is 16kHz. Then ultrasonic motors or ultrasonic wave motors arc named due to 20-100kHz wave motion employed
108
Ultrasonic Motors Technologies and Ap plicalions
for them. The following relationships between above physical quantities exist
]1'= 2rc/w= l/f
II..
(4. ll1)
2rc/k lc=I../1'=w/k =
Figure 4. 21 illustrates the typical propagation of waves in elastic body. The part (a) of the figure only represents spatial state at to, and the part (b) only represents the vibration of point-rD. The phenomena of wave include both parts (a) and (b). Considering a small element in an elastic body, the employing Hooke's law, and the equilibrium conditions of forces, the following wave equation can be obtained:
acp
c' 'V 2 cP
at' where 'V
2
=
-
a + -a + -a' , 2
2
ar'
ay'
az'
and cP
=
(4. ll2)
CP(x,y,z,t).
x
/
(a)
Fig. 4. 21
4.3.2
/
"
"
:'::' 0 :'::' + 1
(b)
/
+~ ~
II(Xo. I )'=Usin(k' o- llll )
~
-U "Z- T----S?
/~
Typical wave propagation in an elastic body
Waves in Elastic Body
In elastic bodies there are following waves:
1. Plane and spherical waves Among solutions CP(.r,y,z,t) of Eq. (1. ll2), the simplest one IS unrelated to both y and z coordinates, and it can be given as follows:
cP
=
(1. ll3)
f(.r- ct)
This wave and the plane perpendicular to the phase, and so it is defined as a plane wave.
.r
aXIs are always m the same
Chapter 1
Fundamentals of Vibration for Ultrasonic Motors
109
1.7:'
In addition, using r = + y2 + Z2 to conduct coordinate transformation for Eq. (4.113), the following solution can be obtained: CP
=
1
(4.114)
- j ( r - ct) r
The same phase wave, which is in the sphere with radius r and the center at the origin, is defined as a spherical wave. In wave propagation process, the locus with the same phase at the same moment is defined as the wave surface or wave front. Wave fronts of the plane and spherical waves arc a plane and sphere surface, respectively.
2. Longitudinal wave When the displacement direction of a wave is parallel to the propagation direction of the wave, which is defined as a longitudinal wave. The general solution of the longitudinal wave satisfying Eq. (4. 112) can be expressed as u(x,t)
=
Usin[2;..IT c.7:±
CIt) ]
(4.115)
where c = c) = IE/pis the propagation velocity of the longitudinal wave in the elastic body, E and p arc the Young's modulus and the mass density of the elastic body, respectively.
3. Shear wave When the displacement direction of a wave is perpendicular to the propagation direction of the wave, which is defined as a shear wave. The general solution of the shear wave satisfying Eq. (4. 112) can be represented as u(x,t)
where c,
=
IG/ p is
=
Usin[2;..IT c.7:±c J ) ]
(4.116)
the propagation velocity of shear wave in the elastic body,
and G is the shear elastic modulus of the elastic body. The shear wave velocity
c,
is about 48% of the longitudinal wave velocity
c)
in
the same elastic body. All waves discussed above are of transmission in an infinite elastic body.
4. Surface wave The surface wave transmits on the free surface of a semi-infinite elastic body. Its amplitude falls exponentially with the depth. The surface wave is similar to the water wave created by a rock thrown in the water. When the earthquake occurs, the motion of the Earth surface recorded by instruments also belongs to the surface wave. Its frequency is very high and its wavelength is very short. Furthermore, it may be a longitudinal wave or shear wave or their combination. The surface wave has two main types: the Rayleigh wave and Love wave. Surface wave ultrasonic motors only involve the Raleigh wave, whose frequeney?:lOMHz(see Chap. 11). Two sets of equation below arc also two solutions for Eq. (4. 112)
Ultrasonic Motors Technologies and Ap plicalions
110
CPl = u(.r,z,t) = U(z)cos(k.r - wt) { U(z) = U(O)C< k' - a'
=
w' I
(1. ll7)
c:
CP2 = w(.r,z,t) = W(z)sin(k.r - wt) { W(z) = W(O)e- PZ k' -
f3'
=
w' I
(1. ll8)
c:
where C z and c z are propagation velocity along the.r and z directions in the elastic body, respectively. Taking the clastic body surface as z = and the depth direction of the clastic body as the positive direction of the coordinate axis (z>O), the above equations show that the wave only exists ncar the surface of the clastic body, which is known as the surface wave or the Rayleigh wave, as shown in Fig. 4. 22. The propagation velocity of the wave with the same phase (wave front) along the .r direction is
°
(1. ll9)
c,=wlk::::::::O.9.j!f
--+
Readers can consult Ref. [14J if they arc interested in the Love wave. _
-0.5
no
Z(w) Z(w)
4.3.3
0'
'.0
U
~~
'\~~+::: \,
Direction of wave propagati on
Fig. 4. 22
U(Z)/U(O) , W(z)/U(O)
I .*l
\\-----$t--
x(u)
Surface wave formed on surface of elastic body
Superposition of Waves
1. Standing wave superposed by traveling waves Two plane waves with the same amplitude but opposite directions in the transmissions are superposed into a standing wave, which can be considered as a special case of two-wave interference. Assume that there arc two plane waves with the same amplitude and frequency: one moves towards right along the x axis, as shown by fine solid line in Fig. 1. 23(a), whereas other travels towards left along the .r axis, as shown by dotted line in Fig. 4. 23 (a). Those arc expressed as following equations
Chapter 1
Fundamentals of Vibration for Ultrasonic Motors
{
Ul = =
U,
Usin(h - wt) Usin(h
111
(1.120)
+ wt)
Superposing these two traveling waves yields :r =
.I1
+ .I2
=
Usin(kI - wt)
+ Usin(kI + wt)
2Usinhcoswt
(1.121)
(a)
(b)
r---~~---r----~----T---~~--~----~~--~ t~~ 4
(c)
Fig. 4. 23
Formation of standing waves
In Eq. (1. 121), the factor cOSwt is related to time. The factor 2U sinkI is the wave amplitude, which is independent of time t and only related to position .I. Whcn t= 0, thc lcft and right traveling wavcs ovcrlap, and thc combincd wavc is shown by coarse solid line in Fig. 1. 23(a). Each point reaches the maximum amplitudc. Aftcr T 14, thcse two traveling wavcs move A/4 towards left and right along the .I axis, respectively, and the amplitude of each point is zero, as shown in Fig. 1. 23(b). After T12, two waves overlap with each other again, and the amplitude of each point reaches the maximum again, as shown in Fig. 1. 23(c). It can be concluded that the above wave obtained by combining two traveling waves makes certain points in the elastic body stationary at any time, whereas the amplitude of other points is twice as much as the amplitude of the traveling wave at any time. This is thc so called standing wavc. In genera!, it is very difficult to generate a traveling wave in the limited elastic body but it is rather easy to yield a standing wave in the same casco Evcn if thc travcling wavc can bc cxcited in the limitcd clastic body, the wave will be reflected from boundary, and the combined wave by the reflected wave with thc initial wavc still is a standing wavc.
4. 3. 4
Formation of Traveling Waves
As noted above, standing waves are actually generated in the limited elastic body, and traveling waves can be gcneratcd undcr following spccial conditions:
1. Traveling wave generated by vibrator and absorber[16-18]. Sashida propos cd a simple mcthod to gcncra te traveling wa vcs, as shown in Fig. 4. 24. It is known from the figure that two actuators (Langevin vibrators)
112
Ultrasonic Motors Technologies and Ap plicalions
are employed: one is employed to produce a vibration and the other to absorb the vibration. In other words. the vibrator in the left end of the elastic body produces vibration which continues to move towards right side, forming the traveling wave. In order to prevent the reflection of the traveling wave from right end of the elastic body, the absorber is applied to consume the energy of the traveling wa ve. This method was used in the linear motor developed by Sashida in 1983: 4 6J • Traveling wave
•
Vibralor
Fig. 4. 24
Absorber
Method of producing traveling waves by vibrator and absorber
2. Traveling waves synthesized by standing waves[47: . A traveling wave can also be excited in a limited elastic body in particular conditions, as shown in Fig. 4. 25. PZT pieces A and B are affixed to a uniform beam simply supported. the voltages cOSwt and cos(wt + a) are applied to PZT pieces A and B. respectively. The corresponding resonance mode (approximate natural mode) is excited at the certain resonance frequency (approximate natural frequency) of the beam. According to Eq. (4. 120). it can be expressed as A-
U1 (x. t)
=
2U 1 sinkxcoswt
=
U 1 sin(kx - wt)
+U
1
sin(kx
+ wt) (4. 122)
B:
u, (x, t)
= =
2U2 sink(x + a) cos(wt + a) U 2sin[k(.:z: + a) - (wt + a) ]
+ U 2 sin[k(.:z: + a) +
(wt
+ a) ] (1.123)
a
Fig. 4. 2S
Excitation of bending traveling wave in a simple supported beam
Taking (k a - a)
=
CPl and (k a
+ a)
=
CP2' Eq. (1. 123) can be changed into
Chapter 1
Fundamentals of Vibration for Ultrasonic Motors
u, (.:c, t)
U, sin(k.:c
=
+ rpl -
wt)
+ U 2sin(k.:c + rp2 + wt)
113
(1.121)
When the elastic body is excited by the PZT pieces A and B, taking the summation of Eqs. (1.122) and (1.121), we obtain a composing wave. Then the conditions exciting such a traveling wave arc
{
rpl rp,
=
ka - a
mrc,
=
m
ka + a = nrc, 2ka = (n+ mh, { a=rc(n-m)/2
=±o, ±2, ±4,'"
n=±1,±3,±5,'" a = A(n+ m)/1
=
Substituting rpl and rp2 into Eq. (4. 124) and combining Eqs. (4. 122) with (4.123) yield u(.:c, t)
+
+
U 1 sin(k.:c - wt) U 1 sine - k.:c wt) + U 2 sin(k.:c mrc - wt) U 2 sin(k.:c nrc wt) = (U j U 2 ) sin(k.:c - wt) (U j - U 2 ) sin(k.:c wt) =
+
+
+ +
+ +
+
(1.125)
When U j = U 2 = U, the traveling wave moving towards right (progressive wave) can be expressed as u(x, t)
2U sin(k.:c - wt)
=
(4.126)
If m is odd and n is even in Eq. (1. 125), the progressive wave vanishes. Only the wave which moves towards left is obtained (retrogressive wave). Therefore, the conditions exciting the traveling wave in the elastic body shown in Fig. 1. 25 become a {a
= =
U1
A(n+m)/1 rc(n-m)/2 = U2 = U
(4.127)
The simplest conditions exciting a traveling wave in a simply supported beam arc
a = A/1, { a = 3A!4, U 1 = U, = U
a a -
rc/2, 3rc/2,
n n
= =
1, 3,
m=O m=O
(1.128)
This is why people have often mentioned that the traveling wave can be formed when the two resonant modes with an identical frequency and mode shapes (standing waves) in an elastic body have 7[/2 phase difference both in space and time. This can be also realized, using multiple PZT pieces divided into two groups, which arc imposed by electric field El and E, , often called as A and B phases, respectively. Let the width of PZT pieces b equal to that of the elastic body, the length be l, and the thickness be hI" and PZT pieces arc placed by accordance with the strain mode shape of the elastic body (referring to the strain mode shape obtained in section 4. 1) where a = A/4, as shown in Fig. 4. 26. The above method of obtaining traveling waves in the limited elastic body is also suitable to circular and ring plates. Moreover, it can be proved that the ring plate's out-of-plane bending displacement and strain mode shapes are similar to corresponding mode shapes of the simply supported beam. Therefore, both the analysis results of the forced vihration of a simply-supported heam and the
Ultrasonic Motors Technologies and Ap plicalions
114
Stmin mode shape excited by A phase
/'
Strain mode shape excited by B phase
_L
Elastic body PZT Piece
Electrode
A
Fig. 4. 26
phase: sill(l)/
B phase: COS(l)(
Sketch of traveling wave in an elastic body by two phase excitation
distribution method of PZT pieces for inducing traveling waves are entirely suitable to the ring stator. Based on the strain modes provided section 1. 1 and the above e.uitation method, it is not difficult to distribute P ZT pieces and to excite two phase resonant modes in order to form a traveling wave. This traveling wave is actually a rotary resonance mode. The traveling wave formation in the circular or ring plates and the best layout of PZT pieces will be further discussed in Chaps. 5 and 6. 3. A traveling wave formed by two phase excitations to a bar[17]. When a fine bar with limited length is in bending vibration, the higher the resonant frequencies are, the closer the two adjacent resonant frequencies will be. Thus. it is possible to form a travcling wave by two phase excitation at the same frequency.
4. 3. 5
Formation of Elliptical Trajectory
It is known from kinematics that if a particle vibrates in two perpendicular directions at the same frequency and certain phase difference. its motion trajectory is an ellipse, i. e. the particle is undergoing an elliptical motion. In the subsequent chapters, discussing the motion principle or driving mechanism of ultrasonic motors, the elliptical motion is always involved regardless whether they are traveling wave type or standing wave type of ultrasonic motors. The horizontal component of the elliptical motion of points on a stator provides the circumferential vcloeity of a rotor, and the perpendicular component provides conditions for imposing pre-pressure. It can be said that the elliptical motion is the precondition of ultrasonic motor's operating. The methods to form elliptical motion include the followings: (1) Elliptical motion of surface wave: when the surface wave transmits in the semi-infinite medium, both its component along the surface and that of perpendicular to the surface can form a typical ellipse trajectory. as shown in Fig. 4. 22. (2) Two traveling waves with certain phase difference. which go backward or forward in two mutually perpendicular directions, can form an elliptical motion (sec Chap. 5). (3) Two standing waves can also form an elliptical motion. For example. the longitudinal and torsional vibration modal response of the stator of a longitudinaltorsional hybrid ultrasonic motor can form elliptical motion trajectories of the
Chapter 1
Fundamentals of Vibration for Ultrasonic Motors
115
points on the stator surface and all points in the stator. In another example, the bending and longitudinal vibration modal response of the linear ultrasonic motor's stator can form elliptical motion trajectories of the points on the stator surface and all points in the stator. (1) In a slim long circular or square bar, utilizing the same modal shape of two bending vibrations on the directions of two principal axes can also form the elliptical motion of the points on the surface of ends.
References [ 1J [2 J
T W Thomson. Theory of Vibration with Applications. Englewood Cliffs New Jersey: Prentice-Hall Inc., 1972. S Timoshenko,D H Young, W J R Weaver,et al. Vibration Problem in Engineering (4'" E-
&. Sons Inc., 1974. Chuanrong Zhou, Chunsheng Zhao. Mechanical Vibration Parameter Identification and Its Application. Beijing: Science Press, 1972. (in Chinese)
dition). "few York: John Wiley
[ 3J [ 4J
Haiyan Hu. Mechanical Vibration and Shock. Beijing: Aviation Industry Press, 1998: 142178. (in Chinese)
[ 5J
Chunsheng Zhao, Fengquan Wang, Weidong Chen. Mechanical Vibration for Engineers. "fanjing: "fanjing Institute o[ Technology Press, 1988. (in Chinese)
[ 6J
Hai Xu, Chunsheng Zhao. Ultrasonic motor linear motor coupled vibration piezoelectric ceramics. China Mechanical Engineering, 2003, 14 (8): 715-717.
[ 7J
Hui Guo, Xijing Han, Chunsheng Zhao. The structure design and material selection o[ the radial drive ultrasonic motors. Journal of Southeast University (Natural Science Edition) , 1999, 29(5B): 80-83. (in Chinese)
[ 8J
Weide Qu, Hcngling Tang, Yiqun Zhang. Mechanical Vibration Handbook (2 nd Edition), Beijing: China Machine Press, 2000: 166-169.
[ 9J
Jian Liu, Chunsheng Zhao. Study on the linear ultrasonic motor based on the vibration in plane of the rectangular plate. Acta Acustica, 2003, 16(S): 86-90. (in Chinese)
[10J
Hai Xu, Weiqing Huang, Chunsheng Zhao. One type o[ linear ultrasonic motor based on the vibration in plane o[ the plate. Journal of Vibration Engineering, 2003, 14(S): 38-40. (in Chinese)
[llJ
Guoxiong CaD. Vibration of Elastic Rectangular Plate. Beijing: China Building Industry Press, 1983: 7-13. (in Chinese)
[I2J
S Rao Singiresu. Vibration of Continued System. Hoboken, "few Jersey: John Wiley &. Sons Inc. , 2007: 471-494.
[13J
R Rog, J R Craiy. Structural Dynamics-An Introduction to Computer Method. New York: John Wiley &. Sons, 1981: 227-229.
[14J
Y Tomikawa. Vibration Theory of Ultrasonic Electronics. Japan: Asakura Bookstore, 1998: 28-53. (in Japanese)
[I5J
Zhcnhua Xiong. Study on Dynamic Characteristics of Piezoelectric Ultrasonic Motors' Stator. Dissertation [or the Degree o[ Master. Nanjing: Nanjing University o[ Aeronautics and
Astronautics, 1998: 12-18. (in Chinese) [I6J [17J
Osamu Nataniguti (Author), Chuanjia Yin (Trans.). Vibration Engineering Encyclopedia. Beijing: Machinery Industry Press, 1983: 453-486. (in Chinese) Hui Guo, Xijing Han, Chunsheng Zhao. The structure design and material selection o[ the radial drive ultrasonic motors. Journal of Southeast University (Natural Science Edition), 1999, 29(5B): 80-83. (in Chinese)
[I8J
Chunshcng Zhao. Development and applications of ultrasonic motor technology. The l" Chi-
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[l9J
Chunsheng Zhao. Ultarsonic motor techniques for 21" century. The Proceeding of 5 d• Small Eletric Motor and Control Technology Symposium. Shanghai, China: N021 Research Institu-
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te of CETE, 2001: 7-12. (in Chinese) Chunsheng Zhao, Hui Guo. Thin-type traveling wave ultrasonic motor. Chinese Invention Patent, ZL01l27037. 3, 2002. Chunsheng Zhao, Guiqing Wang, Long Jin. A new type of self-correction ultrasonic motor using standing wave. IEEE Ultrasonic Sysmposium. Las Vegas, "fevada, USA, 1999: 1-4. (in Chinese) Han Xijing, Guo Hui, Xiaohong Yuan, et al. Study on a new thin type ultrasonic motor. CICEM. Harbin: Harbin Institute of Technology, 1999: 367-370. (in Chinese) Yo shiro Tomikawa, Takehiro Takano, Hiroshi Hirata, et al. An ultrasonic motor using nonaxisymmetric vibration modes o[ a piezo-eeramie annular plate. Japanese] ournal of Applied Physics, 1989,28(1): 161-163. Hui Guo, Heming Sun, Chunsheng Zhao. Several issues in the design of ultrasonic motor using revolving mode in plane. Micromotors, 2000, 33(1): 11-13,36. (in Chinese) Hui Guo, Changqing Liu, Chunsheng Zhao. Study on a novel ultrasonic motor based on inplane vibration mode. The Proceeding of 1'" Small Eletric Motor and Control Technology Symposium. Shanghai, China: No21 Research Institute o[ CETE, 2000: 14-17. (in Chinese) Jian Liu, Chunsheng Zhao. Design of linear ultrasonic motor based on vibration of rectangular plate. New Progress in Vibration and Wave Technology. Shenyang: "fortheastern University Press, 2000: 255-259. (in Chinese) Jian Liu, Hai Xu, Chunsheng Zhao. Study on the linear ultrasonic motor based on the vibration of the thin rectangular plate. Proceedings of the 5 d• International Conference on Vibration Engineering. Nanjing, China: northeasfern University, 2002: 181-189. Zhaoehang Zheng. Mechanical Vibration. Beijing: China Machine Press, 1980: 105-125. (in Chinese) Mingqian Tang, Wenhu Huang, Jun Yun, et al. Manual of Vibration and Shock (2 nd Volume). Beijing: National Defence Industry Press, 1998: 81-106. (in Chinese) Weichang Qian, Kaiyuan Ye. Elastic Mechanics. Beijing: Science Press, 1956. (in Chinese) T Takano, H Hirata, T Tomikawa. Analysis of non-axisymmetric vibration mode piezoelectric annular plate and its application to an ultrasonic motor. IEEE Transaction on Ultrasonic, Ferroelectrics and Frequency Control, 1990, 37(6): 558-565. Hui Guo, Chunsheng Zhao. Finite element analysis of the stator of ultrasonic motor using inplane bending vibration modes. Journal of Vibration, Measurement & Diagnosis, 2000, 20 (1): 236-239. (in Chinese) A Ukita, S Ageha. Hollow cylinder-shaped ultrasonic motor. Journal of Japanese Acoustic Society, 1988,44(3): 173-179. Ye Ji, Chunsheng Zhao. Cylinder type non-contact ultrasonic motor. Journal of Nanjing University of Aeronautics & Astronautics, 2005, 37(6): 690-693. (in Chinese) Ye Ji. Study on the Non-contact Ultrasonic Motor. Dissertation for the Degree of Doctor of Philosophy. "fanjing: "fanjing University o[ Aeronautics and Astronautics, 2006. (in Chinese)
[38J
Ye Ji, Chunsheng Zhao. Modal shape measurement o[ cylindrical stator m non-contact ultrasonic motor. ] ournal of Vibration, Measurement and Diagnosis, 2005, 25 (1): 1-3, 69. (in Chinese) Heming Sun, Chunsheng Zhao, Xiaodong Zhu. Experimental study o[ ultrasonic motor using longitudinal and torsional mode. Small & Special Electrical Machines, 2002 (1): 3-8. (in Chinese) Heming Sun, Chunsheng Zhao, Xiaodong Zhu. Simulation on [rietion characteristic o[
[:l9J
ultrasonic motor using longitudinal and torsional mode. Journal of Southeast University (Natural Science Edition), 2002, 32(4): 624-626. (in Chinese) Heming Sun, Chunsheng Zhao, Xiaodong Zhu. Simulation on friction characteristic of ultra-
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Chapter 1
[10J
[11J
[12J [43J
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Fundamentals 01 Vibration 10r Ultrasonic Motors
117
sonic motor using longitudinal and torsional mode. lournalof Southeast University (Natural Science Edition), 2002, 32( 1): 621-626. (in Chinese) Zheng Tao, Chunsheng Zhao. Some key techniques in design of brush type ultrasonic motor using longitudinal and torsional vibration modes. 1 ournal of Vibration and Shock, 2005, 24 (6): 75-78. (in Chinese) Zheng Tao, Yinhui Dong, Chunsheng Zhao. Research on optimum place of piezoelectric ceramic plate in rod-type ultrasonic motor. Piezoelectrics &. Acoustooptics, 2004, 26 (1): 20-23, 24. (in Chinese) Chunshcng Zhao, Zhirong Li, Wciqing Huang. Optimal design of the stator of a thrcc-DOF ultrasonic motor. Sensors and Actuators A: Physical, 2005 (2): 191-199. Junbiao Liu, Zhihua Chen, Chunsheng Zhao. The driving of a single stator ultrasonic motor with three degrees of freedom and its position control. Small &. Special Electrical Machines, 2003 (2): 11-15. (in Chinese) M Gough, J P G Richards, R P Williams. Vibrations and Waves. New York: John Wiley &. Sons, 1983: 67-75. K F Graff. Wave Motion in Elastic Solids. Columbus: The Ohio State University Press, 1975. T Sashida, T Kenjo. An Introduction to Ultrasonic Motors. London: Oxford Science Publications, 1993. S Ucha, Y Tomikawa. Ultrasonic Motors: Theory and Applications. London: Oxford Science Publications, 1993. Jian Liu, Chunsheng Zhao. Study on the stator of standing wave ultrasonic motor with bend torsion coupler. Applied Mechanics , 1998, 15(S): 159-163. (in Chinese)
Chapter 5
Operating Mechanism and Modeling of Traveling Wave Rotary Ultrasonic Motor A traveling wave rotary ultrasonie motor (TRUM) is presently the most typieal and widely used ultrasonic motor. Its operating mechanism elearly shows energy transforming and its transmitting procedure. and the relationship between a standing wavc and a traveling wavc. which is vcry helpful to understand the motion mechanism of other types of ultrasonic motors. In order to improve the design efficiency of ultrasonic motor and realize precise control, extcnsivc rescarch has been donc on its thcoretical model and pcrformance simulation. Hagood- 1J put forward a comparatively complete analytical model for the first time. After that. Duan:2J. Hagedorn:':. Meiling ZhuC'J. and Xiangdong Zhao L5 - also set up their dynamic models of TR UM. using basically thc samc idca of Hagood. Friend: 5 : and KimCl] studied the dynamic pcrformancc of a piezoelcctric laminatcd circular plate and the motion trajcctory of points on top of thc plate using elassicallaminatcd plate theory. Howcvcr. analytical models used by the researchers mentioned above did not take the structural characteristic dctails into considcration. The shapc of a stator was simplified as an ideal circular plate by ncglecting inner ring and tecth. Thc stators in Refs. [8 J-[ 12 ] were simplified as an Euler or Tiemoshinko beam. which were rather simple for present stator of TRUM. FEM employed in Refs. [13J-[15J consumes much time for parametric analysis. so it is not convenient for performance simulation and structural optimization1l6- 17 - . Hirata1l8 - . Satonobu1l9 - . Uehw 20J • Frayssignes: 21J • Shoushui Wei: 22 23J • et al. simulated the procedure of electromechanical energy transformation using the equivalent circuit model (ECM). Since paramctcrs of cquivalent circuit are detcrmined by tcst rcsults at present. it is difficult to conduct parametric analysis. performance simulation and optimal design. Besides. the contact model is difficult to simulate by ECM. Assuming that there was only slide bctwccn thc stator and rotor. then Coulomb friction law was fulfilled. Ghouti L21 - and Pons- 25 - • et al. set up a contact model of TRUM. in which gcneralized forces on the contact surfacc wcre obtained through an cncrgy method. but its derivate results were so complicated that it is inconvenient for practical application. Wallaschek L26 - and Xiangdong Zhao L27 - made qualitative analysis on the potential stick-slip phcnomenon. considcring thc influence of shear deformation and pointed out the existence of unsymmetrical distributing contact prcssure and scvcral diffcrcnt types of friction loss. Models abovc did not consider the radial slide betwecn the stator and rotor. U eha and Tomikawa: 20J did
C. Zhao, Ultrasonic Motors © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011
Chapter 5
Operating Mechanism and Modeling of Traveling···
119
3-D finite element simulations of the contact area between the stator and rotor of a ring-typc ultrasonic motor, which was uscd in camcra AF lcns produccd by Canon Co., Ltd., but it was too timc-consuming for paramctric analysis. In this chapter, based on the motion mechanism analysis of TRUM, we are going to discuss the electromechanical coupling model of the stator excited by piczoelcctric ccramic picccs- 28J , thc thrcc dimcnsional contact mcchanism bctwccn thc stator and rotor: 29 ] , pcrformancc simulation and paramctric analysis: 30 ] , ctc. The analysis process corresponds to three sub-modules of the motor, namely the stator, contact surface and rotor. Finally the integrated electromechanical coupling model of thc wholc ultrasonic motor is cstablishcd considcring thc structural charactcristics of tccth on stator. Thc model appropriately rcflccts thc incrtial and stiffncss coupling of tccth. Thc influcncc of thc inncr ring on thc dynamic pcrformancc of thc stator is also ineludcd. Also, thc cncrgy loss causcd by radial slide is considered in the contact model. Comparisons of the simulations and test results show that the analytical model in this chapter can reflect the dynamic performance of TRUMs and providcs an analytical mcthod for thc pcrformancc simulation and optimizing dcsign of TRUMs: 31 ].
5.1 5.1.1
Operating Mechanism of TRUM Structure and Operating Mechanism of TRUM
Figurc 5. 1 shows TRUM-60 produccd by PDLab. Its cxplodcd figurc is shown in Fig. 5. 2.
Fig. 5. 1
TRUM-60
Fig. 5. 2
Structure diagram of TRUM-60
In Fig. 5. 2, TRUM is mainly composed of a stator, rotor, shell, bearing, spring, friction lincar, PZT, basc, ctc. Piczoelcctric ccramic is affixcd to stator whilc rotor is affixcd by friction lincr. Friction lincr is bondcd to rotor which contacts with stator through axial pressure.
120
Ultrasonic Motors Technologies and Ap plicalions
Figure 5.3 (a) is the spread vIew along circumference of TRUM. Traveling wave is formed by superposition of two mode responses with equal amplitude and phase difference rr/2 both in time and space, whieh are excited by reverse piezoelectric effect of two groups of piezoelectric ceramics applied with two alternating voltages with equal amplitude, the same frequency and phase shift rr/2 in ultrasonic range. If pre-pressure is applied to the rotor, then the vibration with miero amplitude of points on stator surface will be transformed to rotary motion of the rotor through frictional force.
(a) I
A p hase
B p hase
voltage
voltage
(b) I ~-f
ee) I ~f
Fig. 5. 3
-
Driving mechanism of TRUM
The transmitting procedure of the traveling wave is demonstrated elearly in Fig. 5. 3. When t = 0, point P on stator surface is in state as shown in Fig. 5. 3(a). Sinee the traveling wave propagates right; when t=T/4, the wave peak reaches point P as shown in Fig. 5.3 (b); T/1 later, the traveling wave moves )./4 forward and point P is in state as shown in Fig. 5. 3(c); when t=3T/ 4, the wave valley reaches point P, as shown in Fig. 5. 3(d); when t=T, that is one period later, point P returns to initial state in Fig. 5. 3(a). In a word, when
Chapter 5
Operating Mechanism and Modeling of Traveling···
121
the traveling wave is propagating, circumference posItIOn of point P does not change but its axial displacement and phase keep changing. Figure 5. 3 also intuitively shows that when there is traveling wave propagating to the right in the stator, the motion trajectory of point P is a counterclockwise ellipse. Because of the driving action of points on stator tip, the rotor will rotate in direction reverse to that of the travcling wave propagation. As is known to all, motion is relative, when the stator is fixed (that is why stator is named), the rotor will run because of traveling wave, on the other hand, if the rotor is fixed, the stator will rotate in the opposite direction. It's interesting that reptile in nature bclongs to the latter. Since its little legs move in elliptical motion and the ground can be treated as a fixed rotor, the body of reptile is corresponding to moving stator, as shown in Fig. 5. 1 L32J.
Movement direction Elliptical motion
//-"'----- Stator Rotor
Fig. 5. 4
Similarity of TRUM operating mechanism and crawler crawling
Discussions above briefly illustrate the driving procedure of TRUM. In the next sections, the motion mechanism of TRUM will be studied in detail.
5. 1. 2
Formation of Traveling Wave in Stator
The stator of TRUM is a ring plate of axial symmetry. According to Chap. 4, the natural vibration bending mode of circular plate or thin ring sheet has node circles and node diameters, which arc denoted by B=, where m is the number of node circle and n is the node diameter. In order to get a "pure" traveling wave, two orthogonal" pure" modes (standing wave) with the same frequency and shape on the stator must be induced, as shown in Fig. 5. 5. Functions Bon' which denote two orthogonal modal shapes A and B, (the space phase difference is 90 degrees) can be expressed as
(5. la) (5. 1 b)
where R(r) is the transverse displacement distribution function along the radial direction; sinnIJ and eosn IJ are the displacement distribution functions along the circumferential direction. When the two-phase voltages arc applied on the two-phase piezocleetrie
122
Ultrasonic Motors Technologies and Ap plicalions
(a) A phase modal shape
Fig.
(b) B p ahse modal shape
s. 5 Two orthogonal bending modes with the same frequency and mode shape
ceramic components, without interference of other modes, the two phase A and B modal responses can be written as
WA(r,{},t)
=
W B(r, {}, t)
=
= =
(5. 2a)
WAR(r)sinn{}coswnt W BR( r) cosn{}cos(wnt
+ a)
(5. 2b)
wherc W A and W B arc thc amplitudes of A and B phascs, rcspectively, and a denotes the phase difference between the two phase responses. Thus thc two phase modal coordinatcs arc
qA(t)
=
WAcoswnt
(5. 3a)
qB(t)
=
WBcos(wnt+a)
(5. 3 b)
If the two phase voltages are applied on the two groups of piezoelectric ceramics at the same time, according to the supcrposition principle, thc displacemcnt responsc of the stator is W
=
WA
+w
B
=
~
R(r)[(W A -WBsina)sin(n{}+wnt)
+ (W + WBsina)sin(n{) A
wnt)
+ 2WBcosacosn{}coswnt]
(5. 1)
From Eq. (5. 4), the movemcnt of stator contains a forward traveling wavc sin(n{} - wnt), backward traveling wave sin(n {} + wnt) and a standing wavc cosn {} coswnt. (1) When a=7r/2 and W A = W B = W o ' Eq. (5.4) becomcs thc forward traveling wave (5. 5)
(2) When a
=- ;
and W A
=
W B = W o ' Eq. (5.1) is reduced as (5. 6)
(3) When the above conditions are not met, no "pure" traveling wave will comc into bcing in the stator, which will bc discusscd in dctails in section 5.1. 4.
This shows that conditions for thc traveling wavc to cmerge in thc stator are: when the two e:rciting signals with 7[/2 phase shift in time are imposed on the two groups of specifically polarized Piezoelectric ceramic pieces, respectively,
Chapter 5
Operating Mechanism and Modeling of Traveling···
123
then two phase modal responses on the stator can be acquired with 7[/2 phase shift both in space and time.
5. 1. 3
Elliptical Motion Trajectory of Surface Points on Stator
The ring plate is unfolded into a beam, as shown in Fig. 5. 6, where coordinate x is coinciding with the no-extension and no-contraction neutral axis. According to conclusions in the previous section, when conditions for a traveling wave are met, the wave will come into being on stator. z
x
2h
Movement direction of traveling wave - - - _
Fig.
s. 6
Motion analysis of the surface point of traveling wave
Assuming that the diameter of Stator isrc
'
R(rJ
=
I, Eq. (5.5) can be written as (5.7)
As a micro-deformation beam, the following relationship exists
- -x , erc
(5. 8)
in which, A is wavelength of traveling wave. Combining Eqs. (5. 8) and (5. 7), we get w
=
Wo sin(kx - wnt)
(5. 9)
For a point Po on the surface of the stator with traveling wave movement, when the stator beam deforms, its cross-section will have a rotation angle fl' point Po then moves to point P. Based on the geometric relation in Fig. 5. 6, the displacement of point P in the direction of z and x will be {
Wo sin(kx - wnt) - h( 1 - cosfl)
1;1'
=
Sp
=-
hsinfl
(5.10)
where h is the half thickness of beam. Compared to wavelength A, the displacement of Wo is very small, and fl is also very small, so the displacement of point P along the z and x direction is
(5. 11 a)
124
Ultrasonic Motors Technologies and Ap plicalions
(5. llb) For the tiny beam bending deformation angle f3 (see Fig. 5. 6) can be expressed as dw
f3:::::;:: d.1':
=
(5.12)
Wokcos(k1': - wnt)
Insert Eq. (5. 12)into Eq. (5. 10), we get
Sp :::::;::- Wohkcos(k1': - wnt)
(5.13)
Taking Eqs. (5.13) and (5.11) together, we can get the relationship of the displacement along the z and .1': directions for any point P on the stator surface (5. 11)
Therefore the motion trajectory of points on the stator is an ellipse. The speed along .1': direction of the elliptical motion produces the rotation of the rotor. From derivation of Eq. (5. 13), the speed of point P in the .1': direction can be obtained
_ cit dSI' ,. ( kx _ wnt ) - - Wohkwnsm
V" -
(5.15)
Taking Eq. (5. 15) and considering (5. 8), we get V"o =-hkwnWo
=-h~nWo A
=
V,
(5.16)
which shows: (1) If there is no-sliding between a stator and rotor, V"o is the speed of the rotor, and the minus denotes the rotor velocity is in the opposite direction of that of traveling wave. (2) The rotational speed of the rotor V, is proportional to the axial velocity Wn W o ' thus increasing the axial displacement can advance the rotation speed of the motor. (3) The V, is proportional to axial distance h between point P and neutral layer. Therefore, increasing the height by putting additional teeth on the stator's basement in disk-type traveling wave ultrasonic motor will enlarge the distance between the neutral axis and the stator's surface point so as to achieve a higher rotation speed of the motor. (1) The speed V, is inversely proportional to wavelength A, therefore, increasing modal frequency can reduce the wavelength, the speed can also be enhanced.
5. 1. 4
Effect of Amplitude and Phase on Elliptical Motion
If the formation conditions of a traveling wave in Section 5. 1. 2 can not be satisfied, then the traveling wave excited in the stator will not be pure, which will affect the stability and efficiency of the motor. Obviously, this has to be a voided. In this section, we will discuss the influences of different amplitudes and differ-
Chapter 5
Operating Mechanism and Modeling of Traveling···
125
ent phases (in time) between two orthogonal standing waves on elliptical motion • respectively.
1. Two orthogonal standing waves with different amplitudes Due to uneven material, processing and assembly error and many other reasons. the asymmetry of a stator plate can't be kept. At this time, it will be difficult to ensure the amplitudes of the two phase standing waves to be exactly same. Assuming thatW A =F W B and a = rr/2, Eq. (5.2) will become w
= =
WAR(r)sinnBeoswnt - WBR(r)eosnBsinwnt WAR(r)sin(nB - wnt) (W A - WB)R(r)eosnBsinwnt
+
(5. 17)
When the two standing waves do not have same amplitudes, there will be a traveling wave (the first item) superposed by a standing wave (the second item). At this rate, though the motion trajectory of points on stator surface can be formed similar to an ellipse. it will become tilted and irregular. Fig. 5. 7 shows the numerical simulation results of one point movement within a wavelength A, given different values WW A
=
r; .
B
'1 = 0.9
'1 = 0.6 '1 = 0.5
'1 =0.2
~~\~JO~~\~JO~
'1 =0.1
~\\'~/II~\\'~/II~
1/ =0
I\\"'~//I I \\"'~//II I.
,t
.1
Fig. 5. 7 Motion trajectories of one point on stator's surface using two standing waves with different amplitudes
It can be seen from Fig. 5. 7. when r; < 1. the points have inclined elliptical trajectory, and with a smaller r; we get a smaller Sp and V". When r; = 0, the trajectory becomes an oblique line, which means that SP = O.
2. Two orthogonal standing waves with different phases When two standing waves have same amplitudes (W A = W B = W o ) and different phases, Eq. (5.2) leads to w
= =
WoR(r)sinnBcoswnt + WoR(r)cosnBcos(wnt + a) WoR(r)sin(nB - wnt) WoR(r)cosnB[cos(wnt a)
+
+ + sinwntJ
(5.18)
From the above formula, there arc standing waves and traveling wave in the
126
Ultrasonic Motors Technologies and Ap plicalions
stator at the same time. Moreover the first term is a traveling wave, and the second and third terms are standing waves. The numerical simulation results of the point trajectory are illustrated in Fig. 5. 8.
a=f a= ~
II
~,~ I/~\I/~\~
a=.2..- II 18
I/~\ I/~\I
a=O
Fig. S. 8
I.
,I
A
Motion trajectory of one point on stator's surface
using two standing waves with different phase difference
Figure 5. 8 shows that on the stator surface the point's motion trajectory is an oblique ellipse, and with the phase differences changing from 0 to rr/2, the results are similar to that of Fig. 5. 7 in previous section. From Eq. (5.1), when there are two orthogonal standing waves
III
a stator,
the circumferential velocity of point P on the stator is
(5.19) At a wave crest point,
iJw
ax
IS
equal to zero and () in the above equation can
be eliminated (5.20)
or where R(a)
=
~~~===s=ln=a~======= /sin2 wt
+ sin
2
(wt
+ a)
(5. 21)
Figure 5. 9 shows the value of R(a) with various a value from - rr/2 to rr/2. With a equaling to zero and R(a) equaling to zero, then V" is equal to zero from
Chapter 5
Operating Mechanism and Modeling of Traveling···
127
Eq. (5.20). In this case the phase difference of two orthogonal standing waves in time is zero, i. e. only the standing waves with the same phase in time comes into being in the stator, and the circumferential velocity at the point of wave crest is zero. When a is equal to 1(/2 or -1(/2, g(a) has an opposite sign, which means the corresponding velocities are in the opposite direction. Therefore the speed direction can be altered by adjusting the phase difference.
t···· .... ..... .. ;........t· ..... .. .. , ·,, ... , o ... ..... , .. ...... :.. ................ : ...... .. ....... . ., ', .· . , · .. -0.5 ., .. ., . . . .... , .. ..... .... -I .·, .. ... ..... .. ..... ....... ... . ....... 0.5 ... .. ... ~ ... ... ..
~
, ,
~
~
~
-90
-60
- 30
o
30
60
90
Phase a ngleale)
Fig. 5. 9
Relation of g(a) and phase difference
The analysis above indicates that a pure or perfect traveling wave cannot come into being due to the different amplitude of two standing waves or without 1(/2 phase difference in time. Fig. 5. 10 shows a wave (rotary mode shape) in the sta-
0.2 0. 1
o
- 0.5
-0. 1
-I I
- 0.2
I
1=114
0.2 0. 1
o -0. 1
- 0.2
I
1 =712
Fig. 5. 10
1=3174
Distortion of traveling waves
128
Ultrasonic Motors Technologies and Ap plicalions
tor at the time of 0, 1'/1, 1'/2, and 31'/1 in one period. The amplitudes of the traveling wave are different at different instant points, resulting in unstable operating of TRUM. Moreover all the standing responses eannot be completely transformed into the traveling wave, and the corresponding mechanical energy will dissipate to reduce the output efficiency of TRUM.
5. 1. 5
Polarization Pattern of Piezoelectric Ceramic Components
In order to produce two phase orthogonal standing waves denoted by A and B in space, a piezoelectric ceramic ring is divided into a series of sector areas and usually polarized as shown in Fig. 5. 11. The electrodes for the sector areas are arranged. Positive (+) and negative( - ) signs indicate the polarization directions of sector areas, which are divided into two groups A and B. Between these two groups, there are the sector area of )../1 and the sector area of 3)../1. Moreover the plus polarization sector area of )../1, is called as isolated electrode, which can offer auto frequency tracking signal (see Chap. 13 in detail).
D
rn
c:::::J
No polarization In the positive direction In the negative direction
3AJ4
Fig.
s. 11
Polarization pattcrn of piezoelectric ceramic ring
The two phases of voltages denoted by VA (t) and VB (t) are applied on A and B groups, respectively. In order to correctly depict the voltage applied to the each group on the surface of the ring, the row matrix of voltage sign function cp, can be defined as
cp, ({j) where rpA ({j)
~ {-1,~: 1,
rpB (B)
0, 1,
(5. 22a)
for the A group polarized in the positive direction for the B group or the sectors without polarization for the A group polarized in the negative direction
(5. 22b)
for the B group polarized in the positive direction for the A group or the sectors without polarization for the B group polarized in the negative direction
(5. 22c)
Chapter 5
Operating Mechanism and Modeling of Traveling···
129
The electric potential function is defined as (5.23) where V = [VA (t) VB (t) T are two phases of the alternating voltage imposed on the two groups A and B. To ensure that the driving voltage has the temporal phase rr/2, the two groups of piezoelectric ceramic areas have to be imposed with alternating voltages
(5. 21) In order to obtain a "pure" traveling wave, a "pure" standing ware is necessary. Therefore, for obtaining the latter, in addition to the symmetry of the stator, the teeth number of the stator is best selected as even, and each wavelength must contain the same number of teeth. Because non-polarization and isolated electrode regions in the ring occupies a wavelength region, if the wave number is odd, then A and B groups correspond to even wavelength. As a result, the stator's two responses to A and B groups used for excitations would not interfere each other, so the traveling wave comes into being in the stator.
5. 1. 6
Three-dimensional Motion Analysis of Points on Stator
In Fig. 5. 6, the stator is modeled as a beam and its points undergo a two-dimensional elliptical motion when a traveling wave comes into being in the stator. Actually as the annular thin plate, the points of the stator will undergo a three dimensional elliptical motion, which will be analyzed in this section. Also the dynamic mechanism at the contact interface between the stator and rotor is investigated in order to derive the contact model. When the stator of TRUM vibrates, based on Ritz method the displacement vector of a point in the midplane can be defined as (5.25)
Uo
where U o 'V o ' and Wo are the radial, circumferential and axial displacement component in cylindrical coordinate system respectively. The shape function matrix, CPm' is composed of the n mode shape function vectors. And q is the modal coordinate column matrix. Based on the thin plate theory, it is assumed that no extension and contraction displacement occurs on the midplane, i. e. u,o = 0 and v,o = O. If only two orthogonal bending modes are excited in stator at the same frequency, the modal coordinates and the shape function matrix of Eq. (5.25) are respectively simplified as
q=
[qj
q,J1
(5. 26a)
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Ultrasonic Motors Technologies and Ap plicalions
(5. 26b) where CPwl and CPw2 are mode shape functions, and can be written as CPwl
=
R(r)cos(nB)
(5. 27a)
CPw2
=
R(r) sin(nB)
(5. 27b)
where R (r) is displacement distribution function along radial direction, and cos (nB) and sin (nB) depict the displacement distributions of the two orthogonal modes, respectively. From the integration of Eqs. (5.25), (5.26), and (5.27), the displacement vector of the stator midplane becomes
r 1
=
0
=
R(r)cos(nB)qj (t)
+ R(r)sin(nB)q2 (t)
I (5.28)
Based on the Kirchhoff Assumption, the displacement vector of the stator can be derived as follows (5.29) where u, v, and w denote the components in the r direction (radial), B direction (circumferential) and z direction (axial) respectively in the cylindrical coordinate system; Lmid is the derivation operator matrix, C1Jm is the shape function matrix of the whole stator. We have
L tnid
~ l: C1Jm
0
-zar iJ
1
-
zlr~B
j
(5. 30a)
0 =
LrnidCPrn
(5.30b)
With the stator damp ignored, the modal coordinate vector can be expressed as (5. 31)
where Wo is the modal response amplitude. Substitution of Eq. (5.31) into Eq. (5.29) leads to the radial, circumferential and axial displacement component of any point on the stator as follows: u
=
WoFu(r,z)cos(wnt - nB)
(5. 32a)
v
=
nWaFv(r,z)sin(wnt - nB)
(5. 32b)
w= WoFw(r)cos(wnt - nB)
(5. 32c)
where Fu (r, z) ,Fv (r, z) , and F w (r) depend on the radial, axial, and circumferential coordinates, respectively, and can be written as
Chapter 5
Operating Mechanism and Modeling of Traveling··· CJR(r)
=-z--dr
131
(5. 33a) (5. 33b)
(5. 33e)
By cquation (5.32) a travcling wavc in thc stator will rcsult in thrcc dimcnsional motion of one point on the stator surface, i. e. the three displacement components all exist in the cylindrical coordinate system. We can combine any two cquations in Eq. (5. 32) to obtain thc motion cquations of thc point on cylindrical coordinatc plancs: w'
V'
(nWoF,)'
+ (W oF w)2
1
(5. 31a)
F
(5. 34b)
u=~w
Fw
u' (WoFu)'
+
v' (nW o F v)2
=
(5. 34c)
1
It is found that thc movcmcnt of thc point on thc stator is a thrcc-dimcnsional declining ellipse, as denoted by S in Fig. 5. 12 (a). Thc trajcctory projcction of the ellipse denoted by 1, 2 and 3 respectively in Fig. 5. 12 (b) , which correspond to Eqs. (5. 31a). (5. 31b), and (5. 31c), respectively.
s
0.5 e,
3
0
e. 2
-0.5
(a) Three dimensional Illotion trajectory of driving point on stator surface
Fig. 5. 12
(b) Projection or mo tion trajectory of dri ving point on tator surface
Motion trajectory of driving points on stator surface
The elliptical orbit 1 shown in Fig. 5. 12 and depicted by Eq. (5. 31a), IS neccssary to cnablc thc rotor to rotatc, and is call cd as an cffcctivc clliptical motion: 34 ] in Chap. 7. rcsulting in thc dynamic transmission from thc stator to thc rotor. However the radial component of the point will cause radial slip between the stator and rotor, leading to energy dissipation and efficiency decrease.
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Ultrasonic Motors Technologies and Ap plicalions
With the elliptical motion of the points on the stator surface, the pre-pressure can bc applied on thc rotor. Relativc motion bctwcen thc stator and rotor will crcate a friction force, which is in thc samc dircction as the rotor tangcntial speed. However the tangential speed of points on the stator is in the opposite direction of the friction force acting on the stator. Therefore. the elliptical motion on thc surface points on thc stator and the friction forces at thc interfacc bctwcen thc stator and rotor arc thc nccessary condition for crcating a driving forcc. Morc details of the contact mechanism at the interface are discussed in Section 5. 3.
5.2
Semi-Analytical Electromechanical Coupling Model of Stator
Theoretically the stator of TRUM can be illustrated as shown in Fig. 5. 13, where II and III constitute a piezoelectric laminating plate. Moreover 1, II, III and IV form a completc stator. Inncr web I. substratc II, and tooth IV are madc of thc same metal matcrial. As the transducer, thc piczoelectric componcnt denoted by III will be induced stress and strain, which can be transmitted from the piezoelectric component III to the substrate II by the bonding layer to excite thc standing wavc and thc traveling wavc in the stator. Obviously thc stator with piczoelectric componcnt is a key of elcctromechanical convcrsion of encrgy, which is also different essentially from a conventional electromagnetical motor. Pre-pressure
Fig. 5. 13
Structure diagram of stator and rotor of TRUM
As the piezoelectric laminated plate, the TRUM stator undergoes a coupling action bctwccn mcchanical forcc field and elcctric field. which can bc dcscribcd by the energy method. Based on Hamilton Principle we can obtain the following energy function equation in the stator[os:
f'2 Ldt+ f'2 BWdt
B
t1
=
0
(5.35)
t1
wherc L is Lagrangc function, and BW is the variational work done by thc cxtcrnal forces. In thc following section thc electromcchanical coupling cquation of the stator can be derived by the dynamic substructure method and finite element method (FEM) on thc basis of Eq. (5.35).
Chapter 5
5. 2. 1
Operating Mechanism and Modeling of Traveling···
133
Substructure Division of Stator
The stator of TRUM is divided into three substructures, as illustrated in Fig. 5. 11.
Slibstnlcllire a
Fig.
s. 14
Substructure division of stator
Substructure a: an inner web I; Substructure b: outer annular laminated plate including a substrate II, which is made of the same metal material as the inner web I and piezoceramic ring III; Substructure c: N teeth denoted by IV, which locate circularly on the top of the substructure b. After substructure division of the stator, the modes and dynamic analysis of the substructures can be obtained and the corresponding characteristic matrix (including mass and stiffness matrix) is derived. Then the clectromechanical coupling model of the stator is derived based on the substructural modal synthesis method.
5. 2. 2
Characteristic Matrix of Substructures a and b
1. Displacement field and strain field of a semi-analytical annular element As the axial symmetrical annular structure, substructures a and b can be modelled by a kind of semi-analytical element. The semi-analytical element has two nodes L36 - , and each node has two DOFs- 37J which are an axial displacement and radial rotary angle, Wi and Yi (i = 1, 2), respectively. Based on the model strategy, the stator is discretized as shown Fig. 5. 15. Figure 5. 15 shows that substructures a and b are meshed by two semi-analytical annular elements respectively in the radial direction. Actually more ring elements can be applied to discretize the substructures. Fig. 5. 15 shows the stator is meshed in the radial direction by four semi-analytical ring elements indicated by the numbers j (j = 1, 2, 3, and 1). Each tooth on the top of the stator is discretized by several hexahedrons in the radial direction. It is assumed that e (e = 1,2,3, ... , N) denotes the number of teeth in the circumferential direction and j e (j = 3. 1) represents the hexahedron part of tooth e corresponding to annular element j.
134
Ultrasonic Motors Technologies and Ap plicalions
'r «
,v
"~
III~ ~~
0'
I
SlIbsm,ctur. c
r'-l
• • 2
[fj 3 '
4
Substructure b Substructure a '
(a) Discretized in radiation direction by four ri ng e el me nts
Fig. 5. 15
(b) Stator is divided into several annu lar regions
Discrctized stator in radiation direction by semi-analytical annular clement
When the substrueture is divided into several annular elements, the piezoelectric ceramic component III is meshed by several parts in the radial direction, which correspond to regions 3 and 4 in Fig. 5. 15. The annular areas 3 and 4 of the piezoelectric ceramic component can be considered as the capacitor plate. As the piezoceramic ring has two groups of separated polarization denoted by A and B in the circumferential direction, as shown in Fig. 5. 11, the corresponding charges (see Fig. 5. 11) are defined as QA and QB (j = 3, 1). Then the total charges located on the electrodes of two groups regions can be expressed as Q =
[QA
QB
r.
The following equation can be obtained Q
=
[QA
QBY
=
~QJ= ~ [Q:" j=3
Q~r
(5.36)
j=3
The derivation of Eq. (5. 36) results in 4
I
= Q = ~ Qi
,
=
j-:~
~ j-:-!
[QA
QtiJ
T
(5.37)
The two equations above depict the electrical laws followed by the piezoelectric ceramic components, where I is a current column matrix. According to the plate theory[08: , the displacement function of the annular element can be obtained, as long as the interpolation form of the axial displacement function is given. According to Chap. 4, the nth bending modal response in the axial displacement can be expressed as the following form: 39 ] w(r,B,t) =
w~(r,t)
cosnB
+ w;(r,t)sinnB
(5. 38)
where w~ (r, t) and w; (r, t) are two orthogonal functions that depend on coordinate and time, B is an angle coordinate, cosnB and sinnB depict two orthogonal circumferential displacement distributions of nth modal shape, respectively. The cosine term is called as A phase mode, and the sine one as B phase mode. We have w~(r,t)
(5. 39a)
w;(r,t)
(5. 39b)
where N is the interpolation function matrix in the radial direction, i. e. shape
Chapter 5
Operating Mechanism and Modeling of Traveling···
135
function matrix, which can be denoted as (5.40) where Ni (r) , i expressed as
=
1,2,3,4 consist of Hermite polynomialsc
~) (t)
,
=
[~;.I(t) ~;2
] =
(t)
~
;1
(t)i
Y;j(t)
(5.41)
W;, (t) Y;2(t)
where ~d and ~e2 arc nodal displacement column matrix of the annular elements and correspond to A phase mode. Moreover Wei and Yei (i
=
1,2) denote bending
displacement and corresponding slope at the node. Further,
~'1
and
~'2
are nodal
displacement column matrixes of the annular elements, which correspond to B phase mode. The association of Eq. (S. 38) with Eq. (S. 39) result in the axial displacement f unction of the stator w(r,{},t) =
N~;(t)eosn{}+N~;(t)sinn{}
(5. 42)
The equation above clarifies that the responses of both A and B phase modes arc orthogonal and independent eigenvectors, which simultaneously contribute to the axial displacement. So the axial displacement response of the stator is of the superposition of the two modes when the stator vibrates at nth natural frequency. In this section the shape function is created by the discretization in the local (radial) coordinate, while the analytical method is used for other coordinate directions. This is why the method is called the semi-analytical one. Rearrange Eq. (S. 42) and the following expression is obtained
(5.13) where
[N 1 eosn{} N, eosn{}
N2 eosn{} N4 eosn{}
Nl sinn{} N, sinn{}
N, sinn{} N4 sinn{}]
(S.11a)
(S.44b)
where ~l and ~ i denote the column matrixes of the nodal displacement and its slope vector of nodes 1 and 2 in element j. Taking into account Eq. (S. 43), the displacement field in the semi-analytical annular element can be written as
u
=
[u
v
w]I
=
ClJ,~)
where ClJ, is the shape function of the complete annular element.
(5.15)
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Ultrasonic Motors Technologies and Ap plicalions
(5.16) H ere the deri va tion opera tor is the same as Eq. (5. 30 a) . The relation of the strain field and displacement field of an element in the cylindrical coordinate system can be written as
B=
d ar 1 r a rdB
[0
Eo
Y,o
0
0
a rdB
0
a dr
1 r
[:J ~ L~u
(5. 17)
0
The substitution of Eq. (5.45) into above equation leads to
B,iP
B =
(5. 48)
where the strain tensor B, can be written as - zd-N", 2
dr'
B,
=
L, (/),
_~(dN,,+d2N,,) r
2z r
ar
ra{j
(5.19)
(aN" _ a N,,) 2
rdB
drdB
2. Description of electric field in the piezoceramic ring Usually due to its small thickness (0.3-0. 5mm) , the electric field in the thickness direction of the piezoceramic ring can be considered as a constant
E=
L"gJ
(5.50)
where gJ is the electric potential function.
L,
=
1
h
(5. 51)
p
where hI' is the thickness of the piezoceramic ring. Substitution of Eq. (5.23) into Eq. (5.50) results in E
=
N,V
(5.52)
where Nc is the transformation matrix from the alternate voltages applied to the piezoceramic ring into the electric field intensity (5.53)
3. External forces applied to the semi-analytical annular element Assuming that the charges of A and B groups' electrodes in piezoelectric ceramic area corresponding to the element j is defined as Q J = [Q:" Q~ Y . The variational work due to the charges can be expressed as BW~ =
BQJTV
where V is the applied voltage vector to the piezoelectric ceramic ring.
(5.51)
Chapter 5
Operating Mechanism and Modeling of Traveling···
137
After the stator is divided as substructures, the acting forces at the connection interface between the teeth and element j (j = 3, 4) can be considered as the external forces of the annular element: 41 ] . The nodal force vector from the distributing connection force on the annular element is depicted by Fj{ , where superscript j e denotes the connection forces from the tooth e corresponding with annular element j. In the following section, the substructure interface loading theory and Guyan condensation method will be applied to transform these forces from the connection interface into substructure b. In this way there is no F~ in the electromechanical coupling modeling of the whole stator. More details of substructure interface loading theory can be found in Ref. [42].
4. Characteristic matrix of the annular element Based on Eq. (5.45), kinetic energy of the annular element can be written as TJ
=
~~JTMJ~J 2
(5.55)
where Mi is an element mass matrix and can be expressed as
(5. 56a) (5. 56b) (5. 56c)
where M; and M ~ are respectively the mass matrixes of substrate and piezoelectric ceramic ring in an annular element region, the subscript s only denotes the substrate volume without piezoelectric ceramics, and the subscript p denotes the piezoelectric ceramic volume; V; and V~ represent the integration volume of the substrate and piezoelectric ceramic ring of the element j. The potential energy of a semi-analytical element is defined as
p
=
~LBTC'BdV+ ~LBT(-eTE+C~B)dV ,
(5. 57)
p
where C, and C~ are stiffness matrixes of the substrate and piezoceramic ring, respectively. The first term in Eq. (5. 57) is the strain potential energy due to the mechanical strain of the substrate, the second term (the first term of the second integration formula) is the electric potential energy from the electromechanical effect of the piezoceramic ring and the third term is the elastic potential energy of the piezoceramic ring. Inserting Eq. (5.48) into Eq. (5.57) leads to VJ
=
~~JKJ~J - ~~JnJv 2
2
(5.58)
where K J is the stiffness matrix of the annular element, and n J is the electromechanical coupling coefficient matrix of the piezoceramic ring, which indicates the capability that the structure transforms from electrical energy to mechanical energy.
138
Ultrasonic Motors Technologies and Ap plicalions
(5. 59a) (5. 59b) where K; and K~ are stiffness matrixes of the basic substrate and the piezoelectric ceramic ring of the element j. (5. 60a) (5. 60b) Considering Eq. (5. 52) and the piezoelectric equation in Chap. 2. the electric potential energy of the semi-analytical annular element reads W~
= ~VTcv+~vTnJT~J 2
(5. 61)
2
p
where C p is the capacitance matrix of the piezoelectric ceramic ring and reads
C;,
=
f. vJp
(5. 62)
N;tN,dV
The substitution of the energy expressions above and external forces virtual work expressions in the previous section into Eq. (5. 35) results in the electromechanical coupling equation of the annular element j in the substructure b_ 13 -15J • ,'\r
{
MJ~J +Kj~j
=
c;,v+nJ~J =
nJv+ ~Fi; QJ
J
3.4
(5.63)
The sum in the first equation denotes the total connection forces from N teeth on the top of the stator. and the second equation depicts the charges on the electrode in the annular element region. It is worth of special mention that the substructure a is an inner web without piczoccramics. So for substructure a we have the first equation without the connection forces and no generalized forces from the excitation voltages exist.
5. Characteristic matrixes of substructures a and b Combining the motion equations for the annular elements in substructure a and b and eliminating redundant freedom lead to the following equations M"~"+K"~"=O
(5.64)
(5.65)
where ~' and ~ b are the independent nodal displacement vectors of the substructures a and b; the transformation matrixes C"and C h • describe the boundary dis-
Chapter 5
Operating Mechanism and Modeling of Traveling···
139
placement compatible conditions between the elements of the substructure b. The mass and stiffness matrixes of the substructure a and bare M'
=e
K"
=
l
[Ml
CoT [Kl
M' K2
Je, Je,
Mb Kb
=C [M =C bl
3
M'
bT [K 3
K'
JC JC
b b
(5.66)
After the substructure b is discrete in the radial direction, the bonded piezoelectric ceramic ring can be considered as parallel capacitance. So the charges depicted by Eq. (5.63) will keep to the electric rule of the parallel capacitance as follows
~C:V+ [n'
(5. 67)
j=3
5. 2. 3
j=3
Characteristic Matrix of Substructure c
The substructure c is composed of N teeth in the circumferential direction. Fig. 5. 16 shows one of N teeth is numbered as e (e= 1. 2, ...• N). The tooth is discretized by two isoparametric finite elements 3e and 4e, which belong to the two annular elements 3 and 4, respectively. It is assumed thatKJ' andMJ' are the stiffness and mass matrix of each tooth element, respectively:'6]. Inner web (S ubstructure a)
3
10
4e isoparametric 3e isoparal11e1ric element element
"-y--'
sllbstrate
ceramic
"-y--' Slibstnlcwre b
FEA discretization of tooth e
Fig. 5. 16
Discretized tooth e by two isoparametric clements
Based on the assembly rule of FEM. the mass and stiffness matrixes of tooth e, K' and M'. can be derived by combination of the characteristic matrixes of element 3e and 1e shown in Fig. 5. 16. The nodal vector a' can be expressed as (5. 68) where a~ is the nodal displacement vector, which is called boundary coordinate of joints between the substrate and tooth. such as the node 1, 2. 5, 6. 9, 10 of tooth e, as shown in Fig. 5. 16; the other nodes belong to the inner coordinate, such as the node 3, 4, 7, 8, 11, 12 of tooth e, as shown in Fig. 5. 16. The motion equation of tooth e can be written as
140
Ultrasonic Motors Technologies and Ap plicalions
M~, ] [~~a, MJ[
J+ [K~J Ku
K~, Kh
] [aa~~ ] [0F~h ]+ [F~[0 ] =
(5.69)
where K' and M' are partitioned block forms of the stiffness and mass matrixes, respectively. F;\b is a general force at the interface between tooth bottom and substructure b, and F~[ is a general force at the interface between the tooth top and rotor.
5. 2. 4
Electromechanical Coupling Model of Stator
1. Dynamic effect of teeth on the substrate The characteristic matrixes of the teeth can be transformed to substructure b based on substructure interface loading theory. For any tooth e in Fig. 5. 16, based on Guyan method the inner coordinate can be condensed into the boundary coordinate of the substructure b. (5.70) where 1j/ is the condensed transformation matrix: (5. 71)
where K;, and Kh are partition block forms of the stiffness matrix in Eq. (5. 69) , and Il8Xl8 is a unit matrix. :'\Joting that the displacement vector of six boundary nodes of the e tooth a~ (J = 1,2,5,6,9,10) is the same as those of the joints between the tooth e and substrate II. So the displacement vector of tooth e can be expressed by the generalized coordinate of substructure b: (5.72) where r' is a condensed transformation matrix. Through the loading method:"] at the joints in the dynamic substructure, the mass and stiffness matrixes of tooth e can be condensed into substructure b in the form of Eq. (5.73) {
M'
K'
= =
reTM'r' rTKT'
(5.73)
where M' and K' denote the condensed mass and stiffness matrixes. The dynamic effects of substructure c can be calculated by accumulating the characteristic matrixes of all N teeth in the similar way. The electromechanical coupling equation of substructure b considering approximate dynamic effects of all teeth can be written as (5.71) The boundary forces at the interface become zero because they are from a pair
Chapter 5
Operating Mechanism and Modeling of Traveling···
141
of action and reaction forces formed at the joints between substructures band c. The joints loading theory shows a fact that the teeth affect on the substrate through boundary forces.
2. Motion equations of stator By assembling Eqs. (5. 64) and (5. 74), we can write the matrix equations of substructures a and b in compact form that considers the effect of the teeth. Then considering the boundary condition at joints, the matrix equation of motion of the stator can be given in terms (5.75) where M'
M,
=
T
C,
[
K,
0
=
KC [ C, 0 T
(5.76)
The independent generalized coordinate vector (j, is derived by considering the boundary condition at the joints between substructures a and b. The transformation matrix C, depicts the displacement compatible condition at the joints. We can conduct modal analysis for Eq. (5.75) using a method to eigenvalue problem.
5. 2. 5
Computation Example of Dynamic Characteristics of Stator
TRUM-30 stator is made of phosphor bronze, and its diameter is 30mm. The thicknesses of substructures a and bare 1. 5mm and that of the piezoceramic ring is
o. 5mm.
o.
7mm, respectively, and
There are 15 teeth totally on the top of
the stator with the height of 1. 9mm, and the space between the teeth is
o.
5mm.
Table 5. 1 lists some modal analysis results of the simplified methods in some references, the semi-analytical model discussed in this chapter and testing data measured by Polytec Scanning Vibrometer (PSV-300F-B). Table
s. 1
Modal frequencies of TRUM-30 stator Various modal frequencies/kHz
Various models B04
Boe
B06
B07
Model without the consideration of the inner weh
23. 54
30.62
38.67
47.42
Model without the consideration of the teeth
20. 51
30.98
13.53
57.85
The proposed model in this book
16. 37
21.11
33.03
12.63
Measured results
18.20
25.54
34.86
44.67
Then TRUM-60 stator, made of phosphor bronze and with a diameter of 60mm, is selected for modal experiment. The thicknesses of substructures a and bare o. 7mm and 1. 5mm, respectively, and that of the piezoceramic ring is o. 5mm. There are 72
Ultrasonic Motors Technologies and Ap plicalions
142
teeth totally on the top of the stator with the height of 2mm, the space between the teeth is o. 5mm. The computation and cxpcriment results are shown in Table 5.2. Modal frequency of TRUM-60 stator
Table 5. 2
Various modal frequencies/kHz Various models BOG
B07
BOB
Bog
Model without the consideration of the inner weh
32.63
37. 31
42.89
49.26 44. 10
Model without the consideration of the teeth
22. 56
28.21
35.97
The proposed model in this book
18.11
21.97
30.27
37.98
Measured results
18.85
25.29
32.05
38.20
The first and second rows in Tables 5. 1 and 5. 2 are calculated based on the simplified stator model, which ignores the inner web and teeth. Fig. 5. 17 shows thc frequcncy response curve mcasurcd by PSV-300F-B. The frequcncies corrcsponding to peak rcsponscs arc listcd in thc last line of Tablc 5. 2. Thc comparison indicates that the calculated results of the simplified modes are much different from the measured ones, which means the teeth and inner web have considerablc cffect on thc dynamic characteristics of thc stator. Thc proposed model in this chapter can providc morc accurate results.
ItlJilLJ 10
20
30
40
50
60
I lkHz
Fig. 5. 17
Measured frequency response of stator for TRUM-60
The modal shape of TRUM-60 stator calculated by the proposed model is normalized according to the maximum value. which is similar to the measured results by PSV-300F-B. as shown in Fig. 5. 18. Fig. 5. 19 illustrates thc two orthogonal modc shapcs of B09 calculated by thc proposed model. Through the semi-analytical method the stator is discretized to several elements in the radial direction. and the analytical DOFs as well as the solution spacc increasc. The opcrating of TRUM dcpcnds on two orthogonal mode shapes of the stator excited only under thc special conditions. Assuming that the lincar independent modal vectors of B09 with the frequency Wn are cA and cp" respectively. the corresponding mode shape matrix can be used for the linear transformation of Eq. (5.75)
Ii, wherc q modes.
[ql (t) q, (t)
T
=
f/)q =
[cA
cp, ] [ql (t) ] q, (t)
(5.77)
arc thc modal coordinates of thc two orthogonal
Chapter 5
Operating Mechanism and Modeling of Traveling···
143
I.-------------------~ 0.9 - - Measured rcsults ,," 0.8 Semi-analytical model // 0.7 .§ " 0.6 .- .E .S: 0.5 ;" ./ ~] 0.4 0. '" 0.3 ,;!? ~ ::: g 0.2 . ~ ~ 0. 1
,,
..:.0
O L.o<~-~-~-~-~~----'
0.0 18
0.022
0.026
0.03
Radial coordinate/Ill
Fig. 5. 18
Radial displacement distribution for modal shape
(a) Mode shape (A phase)
Fig. 5. 19
Bog
of stator of TRUM-60
(b) Mode shape (B phase)
Two orthogonal
B09
mode shapes of stator of TRUM-60
Insert Eq. (5.77) into Eq. (5.75) and multiply left by c;pT results in
M,lj+K,q
=
"V+Fe
(5.78)
where Fe is a modal force matrix from the contact interface between the stator and rotor; M, and K, are modal mass and stiffness matrixes; " is a force coefficient matrix. which transforms the electric field vector to modal forces. These can be expressed as (5.79) (5. 80) (5.81) Considering the damping of the stator. the electromechanical coupling equation can be obtained as follows:
(5. 82) Without the contact pressure (i. e. assuming that Fe IS equal to zero). the above equation is decoupled. and it became the two corresponding single-DOF
Ultrasonic Motors Technologies and Ap plicalions
144
systems. For the simulation based on Eq. (5. 82). applying the excitation voltagcs (160V pp ) with thc frcquency of 37. 98kHz. namely. V=[80cos(2rr·
37980t) 80sin(2rr· 37980t)Y. thc stcady-state rcsponse of thc stator will bc a traveling wave with nine wavcs in the circumfcrential dircction. Fig. 5. 20 illustrates the displacement response of a point on the stator surface. X
\0-6
4 .--------------------,,7 . ~,. ------------,
"
3
'-
o
\
\
A,,,jaJ \ /
2
componem
I
\ \
. I \J
\
I
\
"
".
1\
\ CirC<JIllreren,ial ~ co tll l>onetll
I
\
O r---r-----*\----~----~~----~------~
I
-3 -4
I ~
5.26
__
~
__
5.265
"
\
-c~~
5.27
I
I
'-" ____
5.275
I
~
5.28
I
I
I
I
Radial componeru __~__~__~~~
5.285 Time I lislory/s
5.29
5.295
5.3
Fig. 5. 20 Based on proposed method calculated displacement response of a point on thc top of tooth for TRUM-60 for Bog mode
Thc measurcd rcsults of thc stator by PSV shows thc axial displaccment and velocity amplitudes are 3fLm and O. 8m/s. rcspectively. and it validatcs the calculatcd results above. More dctails about expcriments arc prcsented by Chap. 15. From Fig. 5. 20 it is found that there are displacement components of the same order in magnitude in the three directions of the cylindrical coordinate system. The mentioned TRUM-60 stator is chosen as a calculated example. For the stator with the various tooth height but the same other parameters. its Bog modal frequencies are shown in Table 5. 3. Table 5.3
Calculated Bog mode frequencies of TRUM-60 stator at various tooth height
Tooth hcight/mm
0.0
0.5
1.0
1.5
2. 0
Bog modal frequencies/kHz
50.11
47. 12
43.23
41. 76
37.98
The results as listed in Table 5. 3 indicate that the Bog modal frequency of the stator decreases with the increase of the tooth height under keeping other parameters. These results appear more that tooth has considerable effect on the stator dynamic characteristics. which cannot be ignored for the stator motion equation.
5.3
Contact Model Between Stator and Rotor
Making use of the contact interface the vibration of the stator can be converted into the rotary movement of the rotor. Therefore. the power transmission model
Chapter 5
Operating Mechanism and Modeling of Traveling···
145
(the contact model) of the contact interface between the stator and rotor will determine the final output characteristics of TRUM. In this section the forces components acting at the interface and the drive model of the interface are firstly analyzed, then radial friction phenomenon is investigated in-depth. Hereby, the friction loss and power transmission efficiency at the interface are studied in order to simulate the foundation for the performance simulation of TRUM.
5.3.1
Interface Assumptions
The contact interface of TRUM is under the conditions of the vibration in the ultrasonic frequency range and the amplitude could be the level of micron, which make the friction mechanism very complex- 17 -18 - . Therefore, only through reasonable and necessary assumptions, the accurate and useful contact model of the contact interface can be derived. In order to improve the contact characteristics between the stator and rotor, we usually bond a layer of relative soft friction material to the bottom of the rotor (or the top of the stator). Fig. 5. 21 shows that e" ee' and e, are radial, circumferential and axial unit vectors in the cylindrical coordinate system, respectively. Under a certain pre-pressure, due to the relative hard stator and rotor substrate material it can be assumed that there are no contact deformation for the stator and rotor except for the contact layer when ultrasonic motor operates. The rotor
e,t /o'-e
e,"~
z B
____________- .
a
Friction material layer
Fig. s. 21 Losses, transmission efficiency of interface and power for TRUM under various loads
satisfying this hypothesis IS called as a rigid rotor. From Fig. 5. 21, only the friction layer has deformation caused by the contact between the stator and rotor, and its contact region is a ring with small radial width 6.r, r is the central radius. According to the contact theory, the forces acting at the contact interface between the stator and rotor can be decomposed as
in
i,
=
kng
(5. 83a)
=
/l-din
(5. 83b)
where in and i, are the normal and tangential component friction, respectively, k n is the distributed equivalent spring stiffness coefficient of the friction layer, g
146
Ultrasonic Motors Technologies and Ap plicalions
is the normal contact deformation of friction layer, and coefficient of the contact interface.
5. 3. 2
(1d
is the dynamic friction
Interface Force and Power Transmission
Within a wavelength the contact between the stator and rotor IS indicated by Fig. 5. 22, where V T is the moving direction of the traveling wave in the stator, V,{) is the rotating direction of the rotor, and Bv denotes an equal speed point where the contact points on the stator and rotor share the same circumferential speed. In a wavelength, the partieles with the same circumferential speed divide contact area into two different regions of interval: the first type of interval is (- Bo , - Bn) and (B v ' Bo), where the points on the stator teeth surface have smaller circumferential speed than that of the rotor. So the points in the region prevent the rotor rotating, which is denoted by the minus sign; The second region is (- Bv , Bv) , where the points on the teeth surface have larger circumferential speed than that of the rotor. The points in the region push the rotor rotating. Therefore, the compression deformation of the friction material layer can be expressed as g(r,{),z,t) =
w+ h, + h, -
z(t)
(5. 84)
where w is axial displacement, h, is the height from the substrate of the stator top to the midplane of the stator, h, is the tooth height, and z is the distance from the free surface of the friction layer to the midplane of the stator.
-
Rotor substrate
v~
z
I
:
PZT
-------.. VT
I I
I I
_ . .1._._._._._ I
I
p
Fig. 5. 22 Schematic diagram of contact state between stator and rotor in one wavelength
Taking Eq. (5.83) into account results in
in
=
iv
=
{k~~' {1din
(5. 85a)
(5. 85b)
The movement component of the contact points of the stator and rotor, which is projected by er and eo in the coordinate plane, can be illustrated by Fig. 5.23.
Chapter 5
Operating Mechanism and Modeling of Traveling···
14 7
According to Section 5.2. 5, the displacement of the points on the tooth surface has the components with the same order of magnitude in the three directions of the cylindrical coordinates. Obviously, in addition to the axial deformation, the friction layer also has both the radial deformation and circumferential deformation. Therefore, the contact force at the interface is also a three-dimensional vector. In addition to axial component
In
caused by the compression of the friction
layer, the proj ection of friction force I, in coordinate plane formed bye, and eo is not a component along circumferential direction but produces an angle a with the circumferential direction.
Fig. 5. 23
Relative motion analysis at contact interface between stator and rotor
In Fig. 5. 23, Q, represents a contact point on the friction layer of the rotor, V, depicts the circumferential speed on the same layer, which is obtained by multiplying the rotor angular velocity by the rotor radius; point Q, is a corresponding point on tooth e, Va' and V,o are velocity projections along radial and tangential directions, respectively. According to the trend of relative motion between the stator and rotor on the interface, the contact forces acting on the stator and rotor can be identified, as shown in Figs. 5. 24 and 5. 25.
j~
t;
Fig. 5. 24 Contact pressure applied to points on stator surface
In Fig. 5. 24,
In
and
I,
Fig. 5. 25 Contact pressure applied to point on rotor surface
are forces acting on the stator along the axial and tan-
gent direction, respectively.
In is in the negative direction of z, I, is in the coor(Ie> has a positive angle a with
dinate plane formed bye, and eo, and the latter
148
Ultrasonic Motors Technologies and Ap plicalions
eo. fn and f, can be calculated by Eq. (5.83). In Fig. 5. 25 .f~ and f:. are forces (reaction forces of fn and fJ acting on the rotor along the axial and tangent directions. respectively. Thus. the distributivc interface force endured by the stator tceth can be written as (5.86) where f~e and f~, are the absolute values of the circumferential and radial componets of f~. the negative sign ahead .I;, shows that the axial pressure endured by stator is along thc ncgative coordinate axis of z. sgn(V,J. and sgn (V,o - V,) arc sign functions. which can bc dctermincd by thc relativc velocity betwecn the stator and rotor on the contact interface: sgn(V,J
=
1, {+ -1,
sgn(V,o - VJ
=
(5. 87a)
1, {+ -1,
(5. 87b)
The values of f,eand f"for components can be determined by the following formula: { f,o : I f, ~~sa I fer - I f,sma I
(5. 88)
In this formula a is defined as the friction angle which can be dctcrmined by thc relative velocity of the contact point 8 between the stator and rotor
I I tana I = I VI Vcr - V, ,0
(5.89)
Thc friction angle dcscribcs the relativc sliding trcnd along thc radial on thc point 8 bctwcen thc stator and rotor. its rangc is from -180° to 180°. Some references simply considcr that points on the interfacc havc only two-dimcnsional traj ectory formed by the circumferential and the axial motions. i. e. f~ is along direction 8. They is to overlook the radial slip. and the driving cffect is cnlarged.
5. 3. 3
Interface Energy Loss and Power Transmission Efficiency
The frictional power loss on the contact interface can be calculated by the following formula: (5.90) wherc P d, and P dO arc thc power losses causcd by thc radial and circumfercntial slip:
t II f~
~, [1 (
I Vcr I dS) dt
(5.91a)
e-l See)
(5. 91b)
Chapter 5
Operating Mechanism and Modeling of Traveling···
149
Regarding the average energy in a cycle. t is the certain moment when the motor arrives a stable operating. and T is the cycle of the drive voltage imposed on the piezoclectric ceramic components. The transmission efficiency of the contact interface between the stator and rotor can be defined as (5.92) when
Pout
is the mechanical energy output from the motor Pout
1 fr-T . T , M T (3 dt
=
(5.93)
(3 is the rotation speed of the rotor and MT is the load moment endured by the rotor. The transmission efficiency of the contact interface is the indicator which can measure the performance of the interface between the stator and rotor.
5. 3. 4
Contact Model Between Stator and Rotor
Stator teeth and the friction layer are in the state of either contact or isolation. During the course of the contact there are two conditions as shown in Fig. 5. 26: the first is a quasi-contact condition of tooth e. the second is a full contact state of tooth e 1. The radial width 6.r of the contact area is very narrow. and its average radial location is r,.. At this average radius. along the circle. we can sclect a number of auxiliary interpolation points to determine the contact condition between the rotor and teeth.
+
Rotor substrate
e+l
e,
~o o
o Element nodes of tooth • Interpolative nodes of tooth
Tooth e Ca) FE discretization of tooth e and interpolative nodes
Fig. 5. 26
Cb) Contact states of two teeth
Contact state of stator tooth
Fe is a modal force acting on the on stator caused by the distributed contact force of tooth e. After the superposition of the modal forces of n teeth. we can get the total modal force acting Fe in Eq. (5.82) on the stator:
Fe
N
=
~
Fe
(5. 91)
e-j
We can get the axial force acting on the rotor through the integration and su-
Ultrasonic Motors Technologies and Ap plicalions
150
perposition of the axial forces located on the top of each tooth in the contact area. Fi
=
i= Ilfn
(5.95)
dS
e-l SCe)
in which S (e) is the integration on the top of each tooth Similarly, the driving torque acting on the rotor is: MTi
=
t If
In
the contact area.
(5.96)
sgn(V,e - VJrf,e dS
e-l See)
in which the sign function is defined as Eq. (5. 87).
5.3.5
Contact Interface Simulation
TRUM-60 stator, with a diameter of 60 mm, Bog operating mode and made of phosphor bronze, is selected as an example for the following analysis. The rotor is made of duralumin, the excitation voltage and frequency arc 120VI'I' and 37. 98kHz, respectively, and the pre-pressure is 140)J. Fig. 5. 27 shows the axial displacement of the stator and the friction layer compression. Fig. 5. 28 shows the radial or circumferential displacement of the points on the stator surface and the circumferential speed of the corresponding points on the rotor. In this case, (ad' a,2) denote the actual contact area in one wavelength, a01 and a02 arc points with the same circumferential speed of the points on the stator and rotor, and a p corresponds to the wave crest of the traveling wave in the stator. X 10-6
~
S-
~
0
u
~
2
~
0
'0
E
..,"0
Friction layer
J1"
]
-I
.
.
,
'6
]
0
§
jUp
aclj
0
,
.,
10
15
.,
.OJ
Axial
.
i Contact area:
""'" -2 ]. -3 -5
0.2
~
-0
20
" 6
0.6
U
0.8 -5
e"
jac2
25
30
Angle coordinatel(")
Fig. 5. 27 Actual contact area in one wavelength
35
40
,
;I
0.2 0.4
~ ~
,
Contact area
0.4
OJ
'6
0
= " S
~;.
.,,
0.8 0.6
Circumferential
0
10
15
20
25
30
35
40
Angle coordinatel(")
Fig. 5. 28 Velocity components at contact interface in one wavelength
Figure 5. 29 shows the distribution of contact angle in the contact area in one wavelength. Moreover the radial and circumferential contact pressure distributions arc shown in Fig. 5. 30, which denotes that the radial and circumferential components of the contact pressure possess the same order in magnitude. Therefore it can be concluded that the power losses from the radial and circumferential friction also possess the same order in magnitude. It is found that there arc four different ranges of the contact area.
Chapter 5 (1) In (ad'
aOl)
Operating Mechanism and Modeling of Traveling···
151
the points on the stator surface have slower circumferential
velocities than those on the rotor in the negative direction of eo in absolute value. Meanwhile the radial velocities of the particles are along in the positive direction of er • so the contact angle ranges form -180 to - 90 The friction force fro in Eq. (5. 88) is along the reverse direction against the rotary direction of the ro0
0
•
tor. In this case the stator prevents the rotor rotating. I
.
.
Contact area
I
180
7' E
~
~ ?
Q
~
90
.§
"5
or>
.~
0
'6
'" g
U
~
.
Contaci area
I
4 3 2
I
~a - I g -2
- 90
-1 80 -5
.
7 X:..: F I-" O_ ' --~----!----------, 6 - - Circum ferential componenl - Radial compollclll
g -3
a ~l
0
10
15
20
0
25
30
35
40
u -4
-5
0
5
Angle coordinate/CO)
Contact angle distribution in one wavelength
Fig. 5. 29
(2) In (aOl • a p
)
10
15
20
25
30
35
40
Ang le coordinate/CO)
Distribution of contact pressure applied to stator
Fig. 5. 30
the contact angle ranges form -90 0 to 0 0. In this case the ro-
tor is propelled along in the reverse direction against the propagating wave. At the point aOl the stator and rotor possess the same circumferential speed. then they have no relative motion are the eo direction. So the friction pressure is along the radial direction absolutely. and the corresponding serve abrasion happens. (3) In (a p
•
a02) the contact angle ranges from 0 0 to 90 0
•
which denotes that
the radial velocity component of the points on the stator surface become positive. In this case. the circumferential velocities of the points on the stator surface are higher than those on the rotor. So the rotor is still propelled along the reverse direction of the propagating wave. The radial forcing component acting on the stator becomes positive. Point gential contact pressure
t'
aDZ
is the same as point
aOl.
This indicates that tan-
is absolutely along the radial direction. and the serve
abrasion exists in radial one. (4) In (a02 • a c2 ) the points on the stator surface have slower circumferential velocity than those of the rotor in absolute value. and the radial components of the points on the stator are in the negative direction of er • And the contact angle is between 90 0 and 180 0 • In this case the rotor will be prevented by the stator. Figure 5. 31 shows the mechanical power losses at the interface and the output power for various torque levels. In Fig. 5. 31 it is found that the radial friction force causes more dissipated power while no load is applied to TRUM. With the increasing load. the power losses from the circumferential friction become a main part. The power loss of
152
Ultrasonic Motors Technologies and Ap plicalions
30
- - - TOlallosses allhe imerface
25
- -I n the circumferenfia l directi on
-
20
~ 30
"-
~ ; ., ~/
In theradia l direcri on
.--'.- .- .-
15
...... .,"
10
~; /
~
.,.,"
35 30 ::R e::
5
25
".
>.
u
"3:0
4
~ 6
3
5 20 "u E 15
2
10
Co
-
7
6
"
~ 6
O~--~~~~--~--~--~
o
0.2
0.4
0_6
0.8
1.2
OutpUI lorque/(N "Ill )
OUlpUI torque/(N" m)
(a) Lo ses al inlerface under various loads
(b) Transmission efficiency of interface fl , a nd outpul power lmder various loads
Fig. 5. 31 Losses, transmission efficiency and power for TRUM under various loads
5W, which is 60 percent of total losses in the interface, happens under the noload condition. However the power loss is only 2W, which is 10 percent of total losses in the interface when the motor is stalled. Then it is obvious that the power losses from the radial friction cannot be neglected: 49 50J. With the increasing torque, the rotor speed decreases, and the relative motion in the circumferential direction between the stator and rotor at the interface becomes more and more violent. In this case the contact angle becomes small, i. e. the circumferential component of tangential force increases at the interface, and the circumferential friction abrasion is more and more. Meanwhile the small radial component results in the smaller power loss from the radial friction, as shown in Fig. 5. 31( a) .
5.4
5.4.1
Electromechanical Coupling Model of TR UM and Its Simulation Electromechanical Coupling Model of TRUM
1. Dynamic equation of rigid rotor Under the assumptions of the rigid rotor, the rotor moves only have rigid body movement along and around the axis. M, W
+ C~ W =
Fi - Po
(5. 97 a) (5. 97b)
where M, and] , are the mass and rotational inertia of the rotor, respectively, and C~ and C~ are the damping along and around the axis, respectively. They are mainly from the bearings, friction materials, rotor damping material. Fi and MTi are the axial force and driving torque acting on the rotor. Po and MT are the prepressure and load torque acting on rotor, respectively.
Chapter 5
Operating Mechanism and Modeling of Traveling···
153
2. Electromechanical coupling model of ultrasonic motor From the dynamic model of the semi-analytical for stator in Eq. (5. 78) , utilizing the dynamic equations of the rigid rotor Eq. (5. 97) and generalized forces acting on the interface of the stator and rotor Eqs. (5.91), (5.95), and (5.96), we can make the comprehensive electromechanical coupling model of TRUM
t If t II
M,q+C, 4+K,q Mrw+
C~W
=
KV+ Fe
=
fndS- Po
SCe)
Jt~ + c~ /J =
sgn(VrB
(5. 98)
V r ) fre rdS - MT
-
e-l SCe)
The alternating current equation of flowing through the piezoelectric ceramics can be derived from Eq. (5. 67): 1
bc;,V+ KT 4=I
(5.99)
)=3
where 1= [IA III T' is the current column matrix of flowing through the two groups of ceramic electrodes attached to the stator.
3. Energy conversion and output efficiency of ultrasonic motor system The entire electrical energy can be described by the following formula. (5. 100) where Pin is the input power, P oot and P w are the output power and the loss power, respectively: (5. 101) (5. 102)
where t is the certain moment when the motor achieves a stable speed, and Tis the cyele of the alternating voltage imposed on the piezoelectric ceramic compo-
/J
nents, is the rotor rotation speed. P,k is the power loss of the contact interface, and P d, and Pdt arc the power loss due to the damping of the stator and rotor, which arc given as follows, respectively: (5. 103)
1 T
ft.
T
•
c~ (3' dt
(5. 104)
1 ft. T T T t l Vdt
(5. 105)
t
The input power is Pin
=
The output efficiency of TRUM can be defined as (5.106)
Ultrasonic Motors Technologies and Applications
154
5. 4. 2
Performance Simulation of Ultrasonic Motor
1. Mechanical properties Taking the TRUM-60 as an example, based on the proposed modcl the performanee simulation was conducted. All the material and structural parameters, the imposed voltage and pre-pressure are the same as in Section 5. 3. 5. The motor's characteristic curves simulated and tested are shown in Figs. 5. 32 and 5. 33. The signs, are the sampling points from different tests for the same motor, and rough line is the simulation result from dynamic model proposed by the author. The dotted line is simulation results based on simplified contact model ignoring the radial slips between the stator and rotor. As simplified model exaggerates the driving effect, the predicted performance is higher than the testing results[51: Figures 5. 32 and 5. 33 show that 3D contact theory simulation results are quite consistent with the testing results. When the output torque increases to about O. 5:'\J·m, the motor achieves the maximum efficiency, close to 30 percent.
"+"
200 r - - - - - - - - - - - - - - - - ,
- - - Simplified model Proposed model Measured results
175
?
C
125
ioo &
75
•
530
..
Vl
+
+
150
50r-----.-.~_~_-.----------, 45 ." - - - Simplified model .. .... Prorosed model ';f 40 ~. .. Measured results " i; 35 ,/ G
i:§
t
25
~20
8
50
15
,,
25
0.2
0.4
0.6
0.8
1.2
1.4
0.2
Output torque/(N . m)
Mechanical characteristics for TRUM-60
Fig. 5. 32
0.4
0.6
0.8
1.2
1.4
Output torque/(N . m)
Output efficiency characteristics for TRUM-60
Fig. 5. 33
2. The transient characteristics of the motor The transient characteristics of USM refer to the startup and shutdown performance. Its simulation can be done with Eq. (5. 98) , and the results are shown in Fig. 5. 31. The results show that the motor can arrive at a stable operating within only about O. 8ms from the startup. Due to the frictional brake of the interface, the motor's shutdown time is even less, just about O. 6ms to complete stop. The responses of various loads show that motor's transient response time changes a little, only the output speed decreases step by step. 3. Influence of pre-pressure on the output characteristics With the increasing pre-pressure, the contact area between the stator and rotor becomes larger, and then the interface energy loss and power transmission effi-
Chapter 5
Operating Mechanism and Modeling 01 Traveling···
155
ciency will also change. The contact area between the stator and rotor and the compression distribution of TRUM-60 in a wavelength are presented under different pre-pressure as shown in Fig. 5. 35. With the increasing pre-pressure, the contact region becomes larger. Fig. 5. 36
-€
ISO 160
I~
140 120
OAN·m
100
0.6N·m
SO 60
a-
~
is
g 0.6
O.SN-m
13 0.4 .s
0 u" 0.2
0.2 0.4
0.6
1.2 1.6
O.S
0 -5
1.S
lims
3.5
2.5
" g ~ ~
...J
35 100 N 140N
5 4
3
2
... ...
180
-
"" ....
""
0.2
15
20
25
30
35
40
lOON 140 N ISON 200N
30
200N
-
25 20
""
15
" " .... .........
10
1.5
1 0
10
Fig. 5. 35 Contact area and friction layer compression under various pre-pressures
5.5 4.5
0
Angle coordinate in the circumferential directionW)
Fig. 5. 34 Transient characteristics 10r TRUM-60
.2==
1.0
~ O.S
Q)
20
;£ Oi '5 E
•• ··100 N --·140N - --ISO N -200N
.~ 1.2
40
~
1.6
§ 1.4
O.ON·m
0.4
0.6
0 .8 1.0
1.2
1.4
1.6
5L-~--~--~~--~--~~~~
o
0.2
0.4
Output torque/(N -m)
0.6
0 .8 1.0
1.2
O utput torq ue/eN 'm) (b) Total friction losses
(a) Losses from rad ial friction
35,----------------------------, 30
-
25
lOON 140 N ISON 200N
20 15 10
o0~---,'0L..2--c-'0L..4- -0,.....6.,-----0,.....S:---1..L0---I..L.2---I.L.4----'1.6 Oulpu! lorquel(N 'm) (c) Losses from circumferential friction
Fig. 5. 36 Losses at contact interface for TRUM-60 under various pre-pressures and output torques
1.4
1.6
Ultrasonic Motors Technologies and Ap plicalions
156
shows the friction dissipation in the radial and circumferential direction and total friction dissipation in this case of different pre-pressure and output torgue. For the same pre-pressure, there is an extreme value for the power output and efficiency versus output torque, under different pre-pressure. In the operating process of the motor, the pre-pressure is a key factor for the motor characteristic s. Fig. 5. 37 shows load characteristics of TR UM under various pre-pressures.
4. The influence of stator teeth on the mechanical characteristics The stator teeth affect the dynamic nature of the stator, and also affect the output characteristics of the ultrasonic motor. For example, the no-load speed and stall torque of TRUM-60 arc affected by the tooth height under the pre-pressure of 180N. It is noticed that the no-load speed significantly improved with increasing the tooth height, while the stall torque decreases a little. The performance of the ultrasonic motor varies with the different tooth height, as shown in Figs. 5. 38-5. 11. The tangent velocities of the points on the stator surface can be improved with increasing tooth height, then it can increase the speed and improve the output power of ultrasonic motor. At the same time, the radial velocities of the points onthe stator surface can be enlarged. It leads to the higher energy loss from the interface friction between the stator and rotor. Therefore, the efficiency of ultrasonic motor increases with the increased tooth height at first, and then decreases. 6
100 - - 140 N _ . - 180 N 200N
5
~
"0::
Co
5. ;;
50
... \
4
\ \
3
\
0
"'~" ;;;
\ \ \ \ \ \
2
e:: ~
'u Ii:
\
. ..
45 35 30 25
"
20 15
<5
10
g.
100 N - - 140N _ . - 180N 200N
40
,
,
0.2
0.4
0.6
0 .8
1.0
1.2
1.4
0.2
0.4
= E
0::
:;;
. ., ,
ISO
- . . .:.,...
8
f}
C.
;; 0
100 - - 140 N _. - 180 N 2ooN
, ,. '" ......
100
.,
. . ...'
0
~ e::
"'1>-"
'u
15
g.
/
0.2
0.4
0.6
0 .8
1.0
QutPli t torque/(N'Ill)
Fig. S. 37
1.2
1.4
\
\
I:.' (, . t .'
,.
,
'1
10
, \
\
1.0
1.2
1.4
\
\
0
. . . lOON - - 140 N .- 180N 200N
-
, \
\
\
\ \
I
I
<5 0
.. . /
II, ~./J' ~
20
~
50
/~~"
25
ij
~,
0.8
30
...
~ 2oo
0.6
QutP li t torque/eN '111 )
QutPli t torque/eN, Ill) 250
, \
\
\ \
\ \
0
0.2
0.4
0.6
0.8
1.2
Qutp ut torque/eN 'm )
Load characteristics of TRUM-60 under various pre-pressures
1.4
Chapter 5
Operating Mechanism and Modeling of Traveling···
7 c-----------------------------, 6
{
-
Increase of tooth heig ht
5
30 c-----------------------~--,
- 2.0111111 - - 1.5 111111 ••• 1.0 n1l11 - ··0.5m m _ 0.0111111
~g 25 b 20
..• 1.0 mill - · · 0.5 mm _ 0.0 mill
~ 4 o
0-
2.0 mill
- - l.5l11m
157
,;
;; 15
3
e- 2 O
~
10
(ij
5
E
~
0.25
OL--~-~--~-~--~
o
1.25
0.75
0.5
0.25
Output to rqu e /(N 111 )
Fig. 5. 38
0.5
0.75
1.25
Output ot rquel( · m)
Output power
Fig. 5. 39 Losses at the interface at various tooth height
at various tooth height 17 5 r - - - - - - - - - - - - - - , - 2.0 mill - - 1.5 mill ••. 1.0 mill - ··0.5 mm _ O.Omm
35 ,---------------------------__, 2.0 mill - - 1.5mm ••• 1.0 mill -·· 0.5mm _ O.Omm
30 ~ ;, 25
ii" 20 !E ' 13
" 15
25 0L-_~
o
__
0.25
~_~
0.5
__
~_3~
0.75
Output torque/(N' I11)
Fig. 5. 40 Load characteristics at various tooth height
1. 25
0.25
0.5
0.75
1.25
Output torque /(N' I11)
Fig. 5. 41 Output dficiency at various tooth height
The above analysis shows that the tooth height can significantly affect the no-load speed and output efliciency of USM, the actual one in the design must consider users' requirements for the mechanical characteristics.
5. The influence of friction materials Excellent friction materials can decrease the wear at the interface between the stator and rotor and improve the output characteristics of motor. Based on the electromechanical coupling model in this chapter. we can get the trend of mechanical characteristics and the output efficiency of ultrasonic motor versus the different elastic modulus of the friction materials from O. 09GPa to O. 81GPa. According to Figs. 5. 42 and 5. 43, due to the increased elastic modulus of friction layer, the speed, stall torque, output power and efficiency of ultrasonic motor will all increase. It is proved that the appropriate friction materials can not only increase wear resistance and the life of the ultrasonic motor. but also improve the mechanical characteristics. Based on the electromechanical coupling model of TRUM. a performance simulation software is developed, through which we can realize the multi-parameter analysis using friendly interfaces. We can also use it to predict the load characteristics of ultrasonic motors in the design stage, evaluate the merits of their pcr-
158
Ultrasonic Motors Technologies and Ap plicalions
formance and revIse the preliminary design. 250
t
~ 200
0.09G Pa - - - 0. 18 G Pa Increase of elastic modul us - - 0.30GPa _ . . O.84GPa
>l ;;
150
s;.
g.
,
'0
20
"
15
IE 100
6
35 r--------------------------------, Increase of cia ric modulus 0.09 GPa . - - 0,18 GPa 30 '", - - 0.30 GP. ;,R ~ /' ' ,-" 0,84 GPa ::.,. >, 25
~ 6 IO
50
0.3 0.4 0,5 0.6
0. 1 0.2
Output torque/(
0.7
0.8 0 .9
· m)
Fig. 5. 42 Mechanical characteristic vs. clastic modulus
I
//
/'
/'
I'.'
. /. 1/,'
,
.iI,' /'
0. 1
..,,'. ,..
""'",'.
M'
,
0.2
.03 0.4 0,5 0.6 0.7
0.8 0.9
Output ot rquel(N ·m)
Fig. 5. 43 Output efficiency vs. clastic modulus of friction liner
References [ 1
J
N W Hagood. A
J McFarland. Modeling of a piezoelectric rotary ultrasonic motor.
IEEE
Trans. Ultrason. Ferroelec. Freq. Control, 1995, 12(2): 210-221.
[ 2
J
[ 3
J
P Hagedorn, T Sattel, D Speziari, et at. The importance of rotor flexibility in ultrasonic
[ 4
J
traveling wave motors. Srnart Materials and Structures, 1998, 7: 352-368. Meilin Zhu. Study on Piezoelectric Traveling Wave Type Ultrasonic Motor. Post-doctoral Report. Nanjing: Nanjing University o[ Aeronautics and Astronautics, 1996. (in Chinese)
[ 5
J
Jinbo Liu, Xing Ai. Analytical model o[ stator vibration [or traveling wave ultrasonic motor and its driver dcsign. Chinese lournal of Mechanical Engineering, 2001, 10 (0): 129-133. (in Chinese)
[ 6
J
J R Friend, D S Stutts. The dynamics o[ an annular piezoelectric motor stator. lournal of Sound and Vibration, 1997, 204(3): 421-437.
[ 7
J
Y H Kim, S K Ha. Analysis o[ a disk-type stator [or the piezoelectric ultrasonic motor using impcdancc matrix. lournal of Sound and Vibration, 2003, 263(3): 613-663.
[ 8
J
[ 9
J
J unhui H u, Y ongxiao Chen. The study on natured [requency [or ring ultrasonic wave motor stator. Micromotors Servo Technique, 1992, 25(1): 6-10. (in Chincse) S U eha, Y Tomikawa. Ultrasonic Motors Theory and Applications. Clarendon: Oxford Sci-
[lOJ
ence Publication, 1993. Xiangdong Zhao, Chunsheng Zhao. Calculation of natural frcqucneies of the stator with tecth
W H Duan, S T Quck, Q Wang. Frce vibration analysis of piezoelectric coupled thin and thick annular plate. lournalof Sound and Vibration, 2005, 81(1-2): 119-139.
of a tavelling wave type ultrasonic motor. Journal of Vibration, Measure7nent and Diagnosis, 1998, 18(1):213-217. (inChinesc)
[llJ
N lula, Lamberti M Pappalardo. A model for thc thcoretical characterization of thin piczoeeramie rings. IEEE Trans. on [lltrasonics, Ferroelectric", and Frequency Control, 1996, 43
[12J [13J
(3): 370-375. Changliang Xia, Tingna Shi, Yongxiao Chen, et al. Natural [requency o[ the stator o[ ring ultrasonic motor. Small & Special Machines, 1995, 25 (4): 6-10. (in Chinese) Philippe Bouchilloux, Kenji Uehino. Combined finite element analysis-genetic algorithm method [or the design o[ ultrasonic motors. lournal of Intelligent Material Systems and Structures, 200:1, 11 (0):657-667.
Chapter 5 [14J
[I5J [I6J [17J
[I8J [19J [20J [21J [22J
[23J [21J
[25J [26J [27J
[28J
[29J
[30J
[31J
[32J [33J
Operating Mechanism and Modeling 01 Traveling···
159
Changliang Xia. Yao Zheng. Tingna Shi. et al. FEM analysis on stator vibration of traveling wave type contact ultrasonic motor. Proceeding of the CSEE, 2001, 21 (2) : 25-28. (in Chinese) A Frangi, A Corigliano, M Binci, et al. Finite element modeling of a rotating piezoelectric ultrasoniemotor. Ultrasonics, 2005, 13(9):717-755. Chao Chen, Weiqing Huang, Chunsheng Zhao. A review of Modeling for traveling wave type ultrasonic motor. Journal of Vibration Engineering, 2003, 16 (S): 29-31. (in Chinese) Chao Chen, Chunsheng Zhao. Analysis of theory model and optimization design of traveling wave type ultrasonic motor. Mechanical Science and Technology, 2005, 21(12): 1111-1115. (in Chinese) H Hirata, S Ueha. Design of a traveling wave type ultrasonic motor. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 1995, 12(2): 225-231. J Satonobu, "f Torii, K Nakamura, et al. Construction of megatorque hybrid transducer type ultrasonic motor. Jpn. J. Appl. Phys. , 1996, 35: 5038-5041. S U eha, Y Tomikawa. Ultrasonic Motors Theory and Applications. Shanghai: Shanghai Science and Technology Press, 1998. H Frayssignes, R Briot. Traveling wave ultrasonic motor: coupling effects in free stator. Ultrasonics, 2003, 11(2): 89-95. Shoushui Wei, Chuansheng Feng, Qinghua Huang, et al. Simulation and experimental research of equivalent model for ultrasonic motor vibrator. Proceeding of the CSEE, 2003, 23 (10): 125-129. (in Chinese) Qinghua Huang, Shoushui Wei, Chunsheng Zhao. Equivalent circuit model and parameter identification of ultrasonic motor. Micromotors, 2003,36(5): 11-16. "forddin E I Ghouti. Hybrid Modeling of a Traveling Wave Piezoelectric Motor. Dissertation for the Degree of Doctor of Philosophy. Danmark: Department of Control Engineering in Aalborg University, 2000. J L Pons, H Rodriguez, J F Fernandez, et al. Parametrical optimisation of ultrasonic motors. Sensorsandllctuatorsll, 2003, 107:169-182. J Wallasehek. Contact mechanics of piezoelectric ultrasonic motors. Smart Mater. Structure, 1998,7(5): 369-381. Xiangdong Zhao. Study on the Dynamic Modeling and Simulation of the Traveling Wave Type Ultrasonic Motor. Dissertation for the Degree of Doctor of Philosophy. Nanjing: "fanjing University of Aeronautics and Astronautics, 2000. (in Chinese) Chao Chen, Chunsheng Zhao. Modeling of the stator of the traveling wave rotary ultrasonic motor based on substructural modal synthesis method. Journal of Vibration Engineering, 2005, 18(2): 238-212. (in Chinese) Chao Chen, Chunsheng Zhao. Analysis on the dynamic characteristics of contact interface of traveling wave type rotary ultrasonic motors. ] uurnal of Vibration Engineering, 2005,18 (S): 133-137(in Chinese) Chao Chen, Jinsong Zeng, Chunsheng Zhao. Study on the analytical model of the rotary traveling wave type ultrasonic motor. Journal of Vibration and Shock, 2006, 25(2): 129-133. (in Chinese) Chao Chen. Study on Theoretical Model of Traveling Type Rotary Ultrasonic Motor. Dissertation for the Degree of Doctor of Philosophy. Nanjing: Nanjing University of Aeronautics and Astronautics, 2005. (in Chinese) J L Pons. Emerging Actuator Technologies: A Micromechatronie Approach. Chichester: John Wiley &. Sons, Ltd., 2005. Chao Chen, Chunsheng Zhao. Theoretical analysis of the traveling wave rotary ultrasonic motor. The Second International Workshop un Piezuelectric Material and Applications
In
Actuators. Paderborn, Germany, 2005.
[34J
Chao Chen, Chunsheng Zhao. Effect of radial friction in traveling wave type ultrasonic on mechanical performance. The 11"' China Small Motor Technology Conference. Shanghai: No. 21 Research Institute of CETC, 2005: 117-153. (in Chinese)
160
Ultrasonic Motors Technologies and Ap plicalions
[35J [36J
Jiajun Qiu. Analytical Dynamics of Electromechanical System. Beijing: Science Press, 1992. Chao Chen, Zhunsheng Zhao. Modeling of the stator of the traveling wave rotary ultrasonic motor based on semi-analytical method. Chinese Mechanical Engineering, 2005, 16 (21): 1940-1944. (in Chinese)
[37J
Wenliang Wang, Zuorui Du. Structure Vibration and Dynamic Substructure Method. Shanghai: Fudan University Press, 1985. (in Chinese)
[38J
Fubao He, Yapeng Shen. Theory of Plate and Shell. Xi'an: Xi'an Jiao Tong University Press, 1993. (inChinese)
[39J
Chao Chen, Chunsheng Zhao. A novel semi-analytical model of the stator of TRUM based on dynamic substructure method. USE2005. Tokyo, Japan, 2005: 17-25.
[40J
Hongqing Zhang, Ming Wang. Mathematical Theory of Finite Element Method. Beijing: Science Press, 1991. (in Chinese)
[11J
Kailin Jian, Xuegang Ying. Finite element dynamic equation and its element matrices of a rotating beam. ] ournal of Chongqing University (Natural Science Edition), 1988, 21 (1): 49-
[12J [43J [11J [ 45J
[46J [17J [48J [19J
[50J [51J
56. (in Chinese) Yongyan Wang. Theoretical Method and Theory of Dynamic Substructure. Beijing: Science Press, 1999. (inChinese) Haiehang Hu. Natural Vibration Theory of Multi-degrees of Freedom. Beijing: Science Press, 1987. (inChinese) Haichang Hu. Variational Principle and Its Application of Elastic Mechanics. Beijing: Science Press, 1981. (in Chinese) Chao Chen, Chunsheng Zhao. Dynamic analysis of composite stator of ultrasonic motor based on substructure interface loading theory. IEEE ROBIO. Kuming: China Yunnan University, 2006: 331-339. Jiashou Zuo. Finite Element Method of Elastic Mechanics. Beijing: Higher Education Press, 1987. (in Chinese) G M L Gladwell. Contact Problem of Classic Elastic Theory. Beijing: Beijing Institute of Technology Press, 1991. Shu Liu, Jingbo Liu, Ehua Fang. The advances of studies on the dynamic contact problem and its numerical methods. Engineering Mechanics, 1999, 16 (6): 11-28. (in Chinese) Chao Chen, Jinsong Zeng, Chunsheng Zhao. Dynamic model of traveling wave-type rotary ultrasonic motor. Chinese] ournal of Mechanical Engineering, 2006, 42 (12): 76-82. (in Chinese) Yinghui Dong. Study on Rocking Head Type Ultrasonic Motor. Dissertation for the Degree of Master. Nanjing: :'-Ianjing University of Aeronautics and Astronautics, 2003. (in Chinese) Chao Chen, Chunsheng Zhao. Study on the three dimensional contact mechanism of traveling wave type rotary ultrasonic motor. Proceedings of the CSEE, 2005, 26(21): 149-155. (in Chinese)
Chapter 6
Design and Manufacture of Traveling Wave Rotary Ultrasonic Motors With the development of the ultrasonic motor and its increasingly widespread applications, its dcsign and manufacture technologics have been gaining more attention- 1J . In the early period, structure dimensions and materials were often designcd bascd on cxperiences. Prototypes and corrcsponding expcrimcnts were carricd out, with expcrimental rcsults comparcd to the dcsign requiremcnts and adjustments made accordingly-2-. Usually, the whole process takes a lot of time and resources, and design requirements are very difficult to be satisfied. Therefore, it is necessary to find out a general design method to optimize the performance of ultrasonic motorsLHJ. The design of an ultrasonic motor should bc bascd on appropriatc mathematic models, which can predict the performance of the ultrasonic motor and is helpful for its design and manufacture. With the advance in rescarch, thc invcstigation of a theoretical model of ultrasonic motors becomes a focus, just like the early studying of new principles and structures L5 -8J . As shown in Chap. 5, the energy convcrsion in ultrasonic motors include two steps. At thc first stcp, the clectrical encrgy is transform cd into a mechanical vibration of thc stator through the converse piezoelectric effect. At the second step, the mechanical vibration is convertcd into thc motion of thc rotor by thc friction intcrfacc betwecn the stator and rotor. The design method of a traveling wave rotary ultrasonic motor, based on the electromechanical coupling model of Chap. 5, is in this chapter. It should be pointcd out that the dcsign mcthod presented in this chapter is not perfect and nccds to be further improvcd.
6. 1
General Design Process of TR UMs
The general design process of TRUM is the same as that of machines. According to the requirements of clients, the performance parameters of ultrasonic motors are determined, which include the rated torque, rated speed, rated output powcr, sclf-Iock torque, overall structure size, lifc timc, input voltagcs, noisc, etc. Somc clients havc special requiremcnts for ultrasonic motors, such as no clectromagnetic interference, fast response time, hollow structures, applications in the vacuum, etc. According to clicnts' rcquircments for motor performanccs, the material prop-
C. Zhao, Ultrasonic Motors © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011
162
Ultrasonic Motors Technologies and Ap plicalions
erties. structure sizes of various components can be given through the designer's expcricnccs. First thc stator materials and structure sizes should be designed. As a key componcnt. thc stator of TRUM is an annular platc. which is composed of a piezoelectric ceramic pieces and metal elastic material. Two particular modcs with ccrtain frequcncy in ultrasonic range can bc excited in the stator. The optimal design of the stator includes two kinds of special topic. One is optimal dcsign of structurc dynamic characteristics. whosc dcsign objcct function is modal frcquency and modal shapes. and thc othcr is optimal design of structure dynamic response. whose constraints and obj ects is dynamic stress. displacement and so on L9 -. The design of the stator also includes selecting materials. operating modes. dimensions and so on. The stiffness and strength also need to be considcred for thc rotor dcsign. Thc performance of ultrasonic motor can bc predictcd based on the given design parameters of the stator and rotor. According to the design dimensions. thc prototypc are manufacturcd. Then. thc measured pcrformancc of thc motor arc compared with the predicted rcsults. which leads to thc furthcr manufacture or improvement of the design scheme. Fig. 6. 1 shows the design flow of TRUM. which is also applicable to other types of USM.
6. 1. 1
Structure Sizes of the Stator
The structurc sizes of stator include outcr diametcr. mncr diamctcr. substratc thickness. tooth height. tooth number. tooth width. tooth gap. etc.
1. Design of the stator size The dynamic model of TRUM is depicted by Eq. (5. 98) and the output performance has a complex dependence on the design parameters. In the actual design proccss. in ordcr to dctcrmine thc gcneral structure of the ultrasonic motor. designers always want to understand how the outer diameter affects the overall mechanical charactcristics of TRUM. W c can approximatively givc scvcral simplc equations depicting the relationship between the theoretical maximum speed and output torque. output power of TRUM and the outer diameter of stator. The maximum speed of TRUM can be theoretically determined by the circumfcrential spccd of the points on thc stator surface. Whcn the rotor has no relativc movement to the contact point at the stator peak. i. e. the linear velocity of rotor reaches thc circumfercntial spccd amplitudc of thc stator surfacc point.
Thc
maximum specd of TRUM is obtained based on the Eq. (5.16): (6. 1)
where nm"X is the maximum rotary speed of TRUM (r/min). Wo is the axial vibration amplitude of the stator surface point. Wn is the operating frequency and r, is the distance from thc contact point to thc shaft. n is the ordcr numbcr of opcrating mode.
Chaptcr 6
Dcsign and Manufacturc of Traveling Wavc···
163
No
No
No
Technique ~quimnent is
eomplied and prepare to put into production Fig. 6. 1
Design flow of TRUM
Equation (6.1) shows the maximum rotary speed is proportional to the speed factor (wnW o ). As the contact region is generally around the outer edge of the stator, i. e. the maximum speed is inversely proportional to the square of the outer diameter of the stator, as show in Fig. 6. 2 (a). As the radial width of the contact area between the stator and rotor is very narrow, Eq. (5.96) can be simplified as
164
Ultrasonic Motors Technologies and Ap plicalions
i= II
sgn(V,o - V) f,o r' drde """ r; Lr
e - l SCe)
i= f
sgn(V cO
V) f,ode
-
e - l 0( 1': )
(6. 2)
e
where integral interval, (e), depicts angles corresponding to tooth e. It is shown that the torque of motor is proportional to the square of the outer diameter. The substitution of Eq.(6. 2) into Eq.(5. 93) leads to the output power, which is approximately proportional to the square of the diameter. Figure 6. 2 shows the mechanical characteristics of TRUM with different outer diameters. 800
3 2.5
600
.::
~
-S
'§'
E
z
400
"t
.;:
0-
'"
1.5
~
~
200
0.5 0 0.01
0.02
O.oJ
0.04
0.05
00
0.06
0.05
RadiUs/ill (a)
..
~ 3: 0
RadiUs/ill (b)
30
350r---------------------------,
25
300 " \
,
.:
..,E
20
...
'§'
15
250
"
200.....
..... TRUM30 -- TRUM45 -- TRUM60
\ \
'.
\
Co
'"
0-
10
100 50
5
oL-~~~~~--~~--~~
00
0.0 1
0.D2 Radiu (e)
0.03 III
0.04
0.05
o
0.2
0.4
0.6
0.8
1.2
1.4
O ulpu t torque ICN . 111 )
(d)
Fig. 6. 2 Mechanical characteristics of TRUM vs. different stator diameters(for a, b, and c), and speed vs. output torque (for d)
The inner diameter of stator should be moderate. If it is too small, more energy will be transmitted to the inner web of the stator. If it is too large, the substrate ring of the stator will be narrow and the width of the piezoelectric ceramic rings will be smaller. In this case the output power of TRUM is small, and it is difficult to clamp the stator. According to the above analysis, the outer diameter and inner diameter of stator can be initially given.
Chaptcr 6
Dcsign and Manufacturc of Traveling Wavc···
165
2. Design substrate thickness of stator The substrate thiekness of stator has a significant effect on the stator dynamic characteristics, such as the modal frequency and vibration response. Generally speaking, the modal frequency of the stator becomes higher with the increase of the thickness and the stiffness of the stator substrate. In order to avoid noise, the working frequency should be chosen from the ultrasonic frequency band by choosing an appropriate substrate thickness. The substrate should not be very thick to avoid too high working frequencies. Otherwise under the same exciting conditions the corresponding amplitude will be small, which affects the output characteristics of the ultrasonic motor. 3. Design of tooth size According to Eq. (5.15), the circumferential speed of the points on the stator surface can raise with increasing the height of the stator tooth. However, the output torque will decrease if the tooth is too high, as shown in Fig. 5. 11. At the same time, the output efficiency of the ultrasonic motor decreases a little, as shown in Fig. 5. 42. The nonlinear characteristics of the stator will be appeared as well, which could induce a "stall" phenomenon. This should be noticed in the design of an ultrasonic motor. The tooth number and tooth width of stator are interrelated. It is recommended to choose the tooth number as the multiplier of the nodal diameter of the operating mode, and align the lines dividing ceramic elements (sector areas) and with the tooth groove to prevent the inconsistencies of stator's modal frequencies of the two operating modes. The smaller the width of the tooth groove is, the more stable the speed of ultrasonic motor can achieve. By decreasing the width of the tooth groove, larger contact areas between the teeth and the rotor can improve the output performance. Because of the processing limitations, the width of the tooth groove can not be made too small, such as less than o. 3mm, the processing would be difficult. In addition, as the ultrasonic motor is operating, the tooth groove also plays the role of excluding wear debris.
6. 1. 2
Design of Rotor Size
Usually the outer diameter of the rotor is equal to that of the stator in order to ensure that vibration energy of the stator can be fully transmitted to the rotor. The circumferential speed achieves the maximal value at the points on the outer diameter of the stator, where the stator contacts with rotor. In this case the contact area extends, which is helpful for the energy transfered from the stator to the rotor. In the process of the structural design of the rotor, the radial flexible match between the stator and rotor should be considered. In this way the good contact at the interface can be easily achieved and the large local contact stress can be avoided. In addition, the radial friction can be reduced by the flexible rotor to improve the efficiency and output torque. When the motor operates, it is found that the rotor may resonate resulting in the large displacement at the outer edge and
166
Ultrasonic Motors Technologies and Ap plicalions
making the rotor fatigue damage. Therefore the dynamic strength of the rotor should be considered in selecting the structure parameters of the rotor. The displacement response of the rotor can be reduced by bonding the damp materials on the rotor.
6. 1. 3
Choice of Materials
TRUM materials inelude the metal materials, piezoelectric ceramICS, friction material and adhesive materials, etc.
1. Metal material for stator For the stator material of an ultrasonic motor. it IS required to possess such characteristics as wear-resistant, good thermal conductivity, small thermal expansion coefficient (close to the thermal expansion coefficient of the piezoelectric ceramics), light weight. easier machining and low price. etc. In order to meet the above requirements, we have tested some stators made of various materials by various machining methods, as shown in Fig. 6. 3. The following conclusions are obtained: (1) Copper alloy (phosphorous bronze) has good thermal conductivity.
(2) The thermal expansion coefficient of ferronickel (='Ji 36 %. Fe 64 %) IS close to that of piezoelectric ceramics. but has poor thermal conductivity. (3) The powder metallurgy of copper base has good thermal conductivity, best processing and low price. (1) The stator made of wrought aluminum alloy with hardcoat anodize is wearable, lightest in weight, and has moderate thermal conductivity and price level. (5) The stators mode of the other materials shown in Fig. 6. 3 have lower synthetical properties.
Phosphor bronze
Powder metall urb'Y o f coppe r ba e al loy
Aluminum! teel composition
T in bronze wi th sandblast .l'lb. n . ...,
...")U-lLurs
Stainless steel
Aluminum alloy \ ilh nickel pla ting
Fcrronickel
Wro ugh t aluminu m alloy with hardcoat anodize
Illr-lUe U1 UlllereIlL IlIClLeTlcllS anu I-JTUCesslIlg IIleUIUUS
Chaptcr 6
Dcsign and Manufacturc of Traveling Wavc···
167
2. Piezoelectric ceramics As a typc of kcy matcrials, piczoelcctric ccramics havc an important cffcct on thc vibration amplitudc of a stator and thc pcrformancc of USM. Thc piczoelcctric ccramics uscd for cxciting a stator at a high frcqucncy and largc powcr will havc poor pcrformancc which is scnsitivc to tcmpcraturc from dielcctric and mcchanical losscs. So thc piczoelcctric ccramics must havc a high mcchanical quality coefficient Q" a low dielectric loss tanG especially in a strong electric field, a high power density and mechanical strength. In this way a piezoelectric piece can producc high cfficicncy and low hcat. Morcovcr thc piczoelcctric ccramics should have a high piezoelectric constant d 33 and an electromechanical coupling coefficient kp in order to obtain the large torque and the stable operating of USM. However, not all the parameters of piezoelectric ceramics reach optimal featurcs, cspccially for k p , d" , and Qm bcing incompatiblc. Thcy can bc selcctcd by thc USM application fields, which dcpictcd in Chap. 2. 3. Friction materials According to thc thcorctical analysis of Chap. 3, a friction matcrial is onc of thc key materials, which can advance the performance of an ultrasonic motor. In gcncral, selccting friction matcrial uscd for USM includcs an appropriatc friction coefficient, good wear resistance, no friction noise, appropriate hardness, tempcraturc rcsistancc, ctc. So thc friction matcrials should bc dcsigncd as following: (1) good matching friction pair and an appropriatc friction cocfficicnt for thc cxccllcnt pcrformancc of USM; (2) good wcar rcsistancc and a littlc abrasion; (3) no friction noise during the operating of USM; (1) appropriate hardness; (5) closc dynamic and static friction, and no pcristalsis and crccping phcnomcnon during thc opcrating of USM; (6) stcady physical and chcmical charactcristics during thc opcrating of USM; (7) excellent vibration and shock resistance, and low and high temperature re-
sistance.
4. Bonding materials Bonding materials will be applied to USMs m two ways, one IS bonding PZT picccs to a stator substratc, and thc othcr is bonding a friction laycr to a rotor (or a stator). The former requires excellent bonding characteristics and energy transformation cfficicncy. Morcovcr thc good aging rcsistancc is also important. The latter requires excellent bonding and heat dissipation characteristics, and appropriatc contact stiffncss, to obtain bcttcr pcrformancc of USM.
6.2
Operating Modes of Stator and Polarization of PZT Ring
The stator, being a key part of TRUM, whose dynamic characteristics have a vcry important cffcct on thc pcrformancc of TRUM. So thc modal frcqucncy and
168
Ultrasonic Motors Technologies and Ap plicalions
modal shape of an ultrasonic motor must be primarily designed, and then the polarization pattern of piezoelectric ceramics should be given.
6. 2. 1
Design of Modal Frequency
Usually the bending mode B= is applied for TRUM, where m and n indicate the number of nodal eireles and nodal diameters, respectively. It is important to design operating modes(two standing waves), which forms a traveling wave in a stator. The bigger n depicts a higher order mode, which leads to smaller vibration amplitude of the stator at the same excitation. On the contrary, the smaller n indicates a lower order mode. Noteworthily noise in TRUM will be produced if the modal frequency is below 20kHz. Therefore the operating frequency of TRUM lies in the range from 30kHz to 50kHz. For the TRUM stator with the smaller radius than 40 mm, the author has adopted B05 or B07 modes. For that with the larger radius than 60mm, the author has adopted Bog or BOll modes.
6. 2. 2
Polarization of Piezoelectric Ceramics
The stator of TRUM can be simplified as an annular plate, which has infinite modes, as mentioned in Chap. 4. Two orthogonal modes with the same frequency and the same shape are usually selected as the operating modes. This section will discuss the design of a piezoelectric ceramic ring, in order to only excite the operating modes of the stator. In this way the output efficiency of TRUM will be increased. According to the derivation of the annular element matrix in Section 5. 2. 2, the electromechanical coupling matrix presented by Eq. (5. 59b) is an energy transformation matrix. Eq. (5. 53) shows that the transformation matrix is determined by the polarization pattern of a piezoceramic ring. We want to make the electromechanical coupling coefficient as large as possible by the appropriate polarization pattern of the piezoceramic ring in order to excite effectively the operating modes. The bending mode with nine nodal diameters of the annular plate is chosen as operating mode of the stator of TRUM-60. Fig. 5. 11 shows the traditional polarization pattern of a piezoceramic ring, which is proposed by Sashida. In the opinion of the author, the excitation from the piezoeeramie ring polarized in the traditional way is unsymmetrical and it is difficult to excite effectively the symmetrical structure. On the other hand the proposed polarization pattern is shown in Fig. 6. 4(a), in which each polarization unit is 1/4 wavelength instead of 1/2 wavelength in Fig. 5. 11. The applied electric field distribution is shown in Fig. 6.1(b), where the sin, cos, -sin, and -cos voltages are applied to phase A, B, C, and D, respectively. In this way a "pure" traveling wave can be formed in the stator. When an electric field is only applied to single phase of the piezoceramic ring, as shown in Fig. 6. 1 or Fig. 5. 11, the two orthogonal standing waves of the stator come into being. But the two types of polarization pattern have different excitation effect on Bog mode and other modes. Under the same condition the ratio of the response amplitude of Bog mode of the traditional and novel the types of polarization pattern is 1. 7 : 2. 8. It shows that the new kind of polarization pattern ex-
Chaptcr 6
Dcsign and Manufacturc of Traveling Wavc'"
(a) Polar dislribution of PZT ring
Fig. 6. 4
169
DeB A (b) Schemalic diagram of wiring using fo ur-phase exCilalion
Proposed novel polarization pattern of piezoelectric ceramic ring
cites the standing wave (Bog mode) more effectively, whose amplitude increases about 64% as well. When the excitation frequency is elose to that of B08 mode or BOlO mode, the stator based on the traditional polarization pattern will have a considerable response. If the modal frequencies of Bos and BOlO are elose to that of Bog, a mixed mode may be excited. On the other hand the response of interference modes can hardly come into being in the stator based on the new polarization pattern, which avoids effectively the interference of other modes to the operating one. The two piezoceramic rings polarized in two different ways are bonded to two same stators, respectively, to compare their response characteristics. It is found that in addition to Bog many other resonant peaks exist in the frequency response curve of the stator with the peizoceramic ring of the polarization pattern in Fig. 5. 11, as shown in Fig. 6. 5. While only one peak is very dominant in the response curve when the piezoceramie ring with the new polarization pattern III Fig. 6. 4 is applied. Fig. 6. 6 illustrates that the polarization pattern is easier to excite only the operating mode rather than other interference modes.
= E
·0· · · · . .
:
400
~
Novel
'"
..... .
.;1
.
.6
0. E
'"
200
~
'g ~
bd:::J.::::.:::::....::=:::C::~l.~l:::i::===1
o
20
[1kHz
Frequency response of stator for traditional polarization pattcrn
Fig. 6. 5
30
40
50
60
70
1 1kHz
Frequency response of stator for novel polarization pattcrn
Fig. 6. 6
170
Ultrasonic Motors Technologies and Ap plicalions
In addition to the advantages described above, the polarization processing of the new polarization pattern and its polarization effect are much better than that of the traditional polarization pattern. The unidirectional polarization avoids the stress concentration between the two sector areas and reduces the probability of fracture, which often happens in the boundary between the adjacent polarization areas. However the four-phase driver is required to match the piezoeeramie ring with the new polarization pattern:]] 13J. Moreover the insular polar is canceled, so the speed of USM has to be detected by the rotary-type encoder.
6. 3
Structure Form of Stator and Its Modal Analysis
6. 3. 1
Structure Form of Stator
The stator is fixed onto the base with screws, and a thin plate(inner web) is set as a support between the fixed parts and the elastic annular plate, in order to achieve the purposes of vibration isolation in the radial direction. Besides, tooth substructures arc machined on the stator to enlarge the amplitude. The tooth numbers always is an integral multiple of modal order for the symmetry stator structure. Furthermore, teeth should divide circumference in equal parts. Fig. 6. 7 shows the stator structure of TRUM developed by PDLab.
I
I
¢P3 ¢P2 ¢P 1
¢P,
~ r:{' "'-~
I~
Ca) Side view
Fig. 6. 7
Cb) Vertical view
Stator structure of TRUM
In Figure 6. 7, Pl and p, arc the internal and external diameters of the inner web, respectively, P7 is the thickness of the inner web, Pc and PH are used to describe the location of the inner web, Ps is the external diameter of the stator, P3 and p, arc the internal and external diameters of the tooth substructures, respectively, P9 is the height of the teeth, PlO is the width of the tooth groove, and Pll is the tooth number.
6. 3. 2
Modal Analysis of Stator
The shape of the stator is irregular for the introduced structure forms, such as teeth and inner web. For the complex stator structure, the finite element modal analysis can be used to obtain different order modes in the interested frequency
Chaptcr 6
Dcsign and Manufacturc of Traveling Wavc···
171
range, thus to check the existence of desired operating modes and other similar modes near the operating modal frequency. According to the vibration theory in Chap. 4, an annular plate stator has various modes which include both in-plane and out-of-plane modes. If the structure parameters of the stator aren't chosen properly, there will be a lot of modes in a small frequency range, as shown in Fig. 6. 8.
E,,(l n-p lane radi al ex tension mode) 40.17 kHz
Fig. 6. 8
I3oo(Oul- of- plane bending mode) 40.65 kHz
B, s(O ul- of- plane bending mode) 40.95 kHz
Adjacent modes in early design of TRUM-60
Early in the development of TRUM-60 there was the case as shown in Fig. 6. 8, but only Bog mode was the operating mode. The frequency in-plane mode E'1 was 10. 17kHz, lower than that of Bog. At the same time, there was an out-of-plane bending mode B j5 with a node circle,
whose frequency was
10. 95kHz, higher than that of Bog mode. In this case if the driving signal with the frequency close to Bog modal frequency is applied to the piezoceramic ring, the response of the stator includes the contributions of three mixture modes:l<J. As a result, the motor will have an unstable speed and low efficiency. Even using the piezoelectric ceramic ring based on new polarization pattern, the interference modes will be excited if the interference modal frequencies are very closed to the operating mode frequency. Thus the response of the stator is the superposition result of several mode responses and the modal shape is wobbling.
Fig. 6. 9
shows the frequency response characteristics of the stator of TRUM-60 measured by PSV-300F-B.
The modal shapes corresponding to 38 321Hz and
39 023Hz are shown in Fig. 6. 9(a) and Fig. 6. 9(c), respectively. It is obvious that these modal shapes is not pure and interfered by each other. The operating mode, which is shown in Fig. 6. 9 (c), is interfered by the nearby mode shown in Fig. 6. 9(b). What is discussed above is the mode mixture in the near frequency of stator. In fact, high order harmonic components exist when a square wave is used as the driving produced from some drivers, which also induces the mode mixture phenomena in the multiple frequency of the stator. The mode mixture of the stator has bad effect on USM performance. Therefore, the phenomenon of mode mi.Lture should be avoided in the USM design. Even for the manufactured stator, the mode mixture must be checked by experiments. If there is mode mixture, the structural parameters of the stator should be corrected.
172
Ultrasonic Motors Technologies and Ap plicalions
6,-----------~----------------------------------------__.
•
~ E
4
!.
(e) 39 023 Hz
~
(a) 38324Hz
'0 0
~
2 L~_------ (b) 38600Hz
0
38
40
42 1 1kHz
Fig. 6. 9
6.4
Measured frequency response curve of TRUM-60 stator
Sensitivity Analysis and Avoiding of Mode Mixture of Stator
In this section the sensitivities of structure parameters to modal frequencies arc analysed based on FEM analysis of the stator vibration. Then according to the difference of various modal frequencies and structure parameters, we can try to separate the operating modes from other unnecessary ones, which arc called as the interference modes. In this way the mode mixture of the stator can be avoided.
6. 4. 1
Principle of Sensitivity Analysis
Assuming that Wk is the kth modal frequency of stator and Pj is the jth design parameters, the first order sensitivity of the kth modal frequency of stator with respect to p; is defined as:]5] dWk d P;
,
j
=
1,2,···,N
(6. 3)
The first order rela ti ve sensi ti vi ty is defined as dWk Wk dp;
P;dWk WkdP;
=
=
(6. 1)
P; According to the materials and dimensions of the stator, the stiffness matrix of stator K and the mass matrix M can be obtained by FE software (A:'\ISYS). The characteristic equation can be written as KCPi-W'MCPi
=
0,
I
=
1,2,3,···
(6. 5)
Eigenvalues and eigenvectors can be calculated by solving Eq. (6. 5). Therefore, modal frequencies and shapes of the stator can be obtained. The operating modal parameters of the stator, Wk andcpk' satisfy Eq. (6. 5) as follows KCPk - W;MCPk
=
0
(6. 6)
Chaptcr 6
Dcsign and Manufacturc of Traveling Wavc···
173
where Wk • I/>k • K • and M are the functions of design parameters Pl. Therefore. the partial derivative of the Eq. (6.6) can be written as[15:
uK _ 2w M UWk, - w' UM,] A, [ dPj k dP 1 kdP1 'f"k
+ (K -
w' M) dl/>k, k uP1
=
0
(6.7)
Multiplying Eq. (6. 7) on the left by I/>Y leads to (6. 8)
The mass matrix and stiffness matrix is symmetry. so we can obtain
I/>Y(K-w;M) =I/>Y(K-w;M)T
[(K-w;M)l/>kJ T
=
=
0
(6. 9)
Therefore. Eq. (6.9) can be written as ".;r
'f"k
[dK, _ 2 dM, ] A, UP1 Wk UP1 'f"k
_
uWk, A,T A, 2Wk d P1'f"k M 'f"k
=
0
(6. 10)
The sensitivity of the kth modal frequency Wk to structure parameter P1 is defined as
UWk dP 1
=
1 A,T [dK , dM] A, 2Wkl/>l'Ml/>k 'f"k uP1 -Wk UP1 'f"k
(6. 11)
According to the second kind of mode normalized method mentioned in Chap. 1. we can obtain (6.12)
1 Therefore. Eq. (6.10) becomes
UWk __1_ A,T [dK _ 'Yk d P1 2Wk UP1
2
Wk
dM] A, 'Yk UP1
(6. 13)
The relative sensitivity of modal frequency to structure parameter is
S1< ,
"- P
=
P1UWk Wkdpj
=
AA,T 2 'Yk
2Wk
[dK _ '1 uP 1
, dM] A, uP1 'Yk
Wk '1
(6. 11)
Based on FE analysis. it is difficult to describe the mass matrix M and stiffness matrix K of discrete structure as a continuous function of design variables. So. Eq. (6.11) can be solved by using difference approximation and perturbation method. When parameter P1 has a perturbation LP1
•
the matrix M and K will
have perturbation LM and 6,K. respectively. Therefore. we can obtain (6. 15)
6. 4. 2
Sensitivity Analysis of Stator for TRUM-60
Table 6. 1 shows the structure parameters of some TRUM-60 stator developed by PDLab. which arc illustrated in Fig. 6. 4. The material of the stator is phosphor bronze.
174
Ultrasonic Motors Technologies and Ap plicalions Initial size of TRUM-60 stator(Unit: mm)
Table 6. 1
PI
P,
P3
p,
Pc
32
44
44
60
60
P7
ps
po
PIO
PI I (Tooth number)
0.7
0.6
1.5
0.5
72
Pc
Based on the strueture parameters in Table 6. 1. the stator FE modcl (3 168 nodes and 1 584 clements) is built. When the parameters have perturbations (increased by 1%), it can be obtained to calculate the sensitivities of the operating modes Bog and other adjacent modes B15 • which are out-of-plane bending modes with one nodal circle and five nodal radiuses, and E21 , an in-plane extension-contraction mode. as shown in Fig. 6. 10. Although the stator has many types of modes. for each type of modes its mode frequencies increases with the mode order. It means that the frequency sensitivity of the mode of some type to design variables can represent that of this type of modes. According to the sensitivities of different types of modes. it is possible to fulfill mode separation. 2
x 10"'
I
1.5 I
I
I
•
t
:
:
I
- - - - - ~ - - - - - -:- - - - -- +-- - - - -:- - - _. --i - - . 1/-+£&&,1~l , , ._ J ______ , ____ _ ----- 1. ------'-----"-, , , - -- , I ____
-----
x:
~
I
~
;-- n-----
f --- ' -----~ ~- --- · ----
:,
- 0.5 - - - - -
I
_ 1I
I _
_
_
..L _ _ _ _ _ I
I _ 1_
_
_
_
0.
_
I
I
_ __ _ '- _ _
_
__
J _ __ _ __
I L
_
_ _
_
_
, :
____- L_ _ _ _ p, p, p,
~
lia '
__= r__~~__rm~~Ub~~~~~~--~ I~I ~ ~ ~ : ~ :' :,L..J
1 :,, ;;;1 -1,1
O ~--~~_,
-1 L-__
_ 8.,. c::::::::J B"
I
_ L_ _ _ _~_ _ _ _L __ _ _ _L __ _~_ _ _ _~
p,
p,
p,
p"
Design variables
Fig. 6. 10 Sensitivity of Bog mode and its adjacent modes of TRUM-60 to structure parameters
Figure 6. 10 shows that the sensitivity of Bog to parameter PI is very small and that of B15 and E'l to P1 is relatively bigger. Therefore, it is possible to separate Bog from B 15 and E'I by adjusting PI. The sensitivities of E'I to structure parameters Pc , P8' and PlO are contrary to that of B09 and B15 • So, it is possible to separate in-plane extension contraction mode E21 from others.
6. 4. 3
Mode Separation of Stator for TRUM-60
In order to validate above method. the stator of TRUM-60 shown in Fig. 6. 9 is chosen as the example of the mode separation. The structural sizes of the stator are given in Table 6. 1. The frequency of the operating modes and other adjacent modes are calculated and are shown in Table 6. 2. It is noticed that the frequency
Chapter 6
175
Design and Manufacture of Traveling Wave···
of the operating modes Bog are close to those of other modes, i. e. 113Hz higher than that of B I5 and 812 Hz higher than that of E 21 • In this case the modal mixture exists. It is possible to separate the operating modes from the non-operating modes by reducing the frequencies. According to the sensitivity analysis in Fig. 6. 10, the decrease of PI can reduce the modal frequency of B15 and E21 , which results in the mode separation expected. When PI is 28mm. the calculated modal frequencies closing to Bog are shown in Table 6. 3. There is the difference of 2741Hz between Bog and B15 after PI is adjusted. In practice, it is found that the normal operating of an ultrasonic motor can be hardly affected by the interference modes with the frequency difference of 2kHz at least from operating modes. As a result, the modified stator can meet the design requirement. Table 6.4 shows the measured modal frequency of the stator of TRUM-60 before and after the mode separation. Table 6.2
Calculated modal frequency of stator for TRUM-60 before mode separation
Various types of modes Modal frequency/Hz
Table 6.3
38 243
38 612
39 055
44 325
Calculated modal frequency of stator for TRUM-60 after mode separation
Various types of modes Modal frequency/Hz
Table 6.4
35 784
36 229
38 970
42 764
Measured modal frequency of stator for TRUM-60 Various types of modes
pdmm E 21 /Hz
Bog/Hz
Before mode separation 32
38 167
38 321
39 023
13 856
After mode separation 28
35 413
35 824
38 660
41 367
10
~E ~
'u 0
5
~
0
36
38 j lkHz
40
Fig. 6. 11 Measured frequency response and mode shapes of TRUM-60 after mode separation
176
Ultrasonic Motors Technologies and Ap plicalions
Based on the adjusted sizes. another stator is manufactured. The out-of-plane bending modes are observed by PSV-300F-B. The results are illustrated in Fig. 6. 11. which shows the peak and modal shape of out-of-plane bending mode B 15 • because only out-of-plane modes can be measured by PSV-300F-B.
6.5
Optimal Design of Stator
After determining the structure form of a stator. material properties and operating modes arc given. the parameter optimization of the stator structure is followed. In this section an optimization model is presented from the design requirements of the stator. taking into account the nature dynamic and dynamical response. Based on parametrieal finite element analysis in Section 6. 3 and sensitivity analyses in Section 6. 4. the sequential quadratic programming is adopted1l7 -
6. 5. 1
Optimal Model of Stator
Parameter optimization of a stator is discussed in this section. and design parameters are continuous in variation ranges. After the parametrieal finite element model in Section 6. 3 is applied. and the optimal design of the stator can be considered as the mathematical model of the corresponding optimization. In this way the optimization of the stator parameters is conducted with the given ultrasonic frequency. operating mode. appropriate difference between the operating mode and others and the special operating ranges. The optimal model is mathematically written as follows max s. t.
obj=Wov" p:h~Pi~P:'h
(6.16)
Lf~2kHz
fo~20kHz
where VB is velocity amplitude in the circumferential direction. p~h and p;'h are the upper and lower bounds of variation range of the ith structure parameter. respectively. Lf is the frequency difference between the operating modal frequency and the nearest interferential modal frequency. fa is the operating modal frequency. While some stator parameters are fixed. the others arc variable. namely design variables. Design variables arc constrained by factors of the structure form and processing technique. such as the minimum of the tooth groove param-
eter PlO is o. 3mm. Based on the above mentioned. the first elass constraints are formed. namely boundary constraints. The second elass constraints are formed by requirements of dynamic perform-
anees. namely performance constraints. The two constraints arc derived from enough large frequency differenee(Lf >2kHz) and operating modes in the ultrasonic frequency range. The relationships between stator parameters and dynamic performances obtained by finite element analysis are nonlinear implicit functions.
Chaptcr 6
Dcsign and Manufacturc of Traveling Wavc···
177
In Eq. (6.16) the dynamical response performances are regarded as the object. The limit speed of an ultrasonic motor is theoretically derived from the circumferential speed of the points on the stator surface- 18J . The speed factor wnWO (product of operating modal frequency and axial amplitude) is regarded as the evaluation index119-zoJ. According to the simulation analysis in Section 5. 1. 3, the output characteristics of an ultrasonic motor are greatly related to axial amplitude and circumferential velocity. The product of axial amplitude and circumferential velocity Wo v" can be regarded as another evaluation index: 6: , which can be transformed to wnW;. Compared with speed factor, the index much more emphasizes the contribution of axial amplitude to the output characteristics of USM. So the latter index is adopted here. The axial displacement and circumferential velocity can be gained from the harmonic response analysis under the conditions of the certain operating modal frequency fa and excitation voltage.
6. 5. 2
Example of Optimal Design of Stator
The exemplification of optimization design of TRUM-40 stator IS gIven here. The parameters of a piezoelectric ceramic ring are known, the inner and outer radiuses are 28mm and 40mm, respectively, and the width is o. 5mm. Firstly, the B07 modes are adopted as the operating modes based on trial calculation. According to the similar parameters of TRUM-60 stator and processing technique, the parameters of TRUM-40 stator are defined preliminarily. The five parameters Pz, P3' Pl' P5' and Pll are regarded as fixed and listed in Table 6. 5, the others as design variables. Table 6.5
pz 28
P3 28
The fixed parameters of stator for TRUM-40 (Unit:mm) Pl1 (Tooth number) Pc 40
40
56
The initial values of p, which reads [ Pl ,Pc, P7 ,p, ,P9 ,PlO J, is [20 o. 5 o. 5 5 2.0 o. 5J, and the upper and lower boundaries of variation range of the design variables plb = [11 o. 5 o. 5 o. 5 1. 0 o. 1J, pub = [21 2. 5 1. 0 2. 5 3. 0 o. 8].
o.
The mathematical form is as follows mm s. t.
obj=-Wov,,= f(pJ cj (pJ = Pi- p~b~O(i= 1,6,7,8,9,10; j= 1,2, ... ,6) C5
C ll C1
(pJ = p;'b - Pi~O (pJ = /:::,.f- 2~0 (Pi) = fa - 20~0
. }
C10
,(Pi)=50- fo~O (6. 17)
The optimization problem is solved by a numerical iterative search algorithm, namely sequential quadratic programming (SQP) algorithm. The sensitivity analysis method in Section 6. 3 is used to compute sensitivities of the operating modal frequency and frequency difference, and the forward difference method is used for the sensitivity of the evaluation index, the difference step is one percent
178
Ultrasonic Motors Technologies and Ap plicalions
of current values. The search direction is solved by the quadratic programmmg sub-problem formed by the sensitivity analysis results. and the search step is solved along the solved direction by unconstrained problem, namely min obj1 =
-Wov" +r[2.:
1min{O.cJ(p)} IJ.
here the penalty factor r is 1 000. obj1 is
the sum of the object and penalty term. The convergence eriterions of dot pitch (2-norm distance) and the decrease of the objective function arc adopted, and the values of termination accuracy are 1. Oe-3. This algorithm converges within 31 steps, and Fig. 6. 12 shows the change process of the object and penalty term.
.,o ""
'vvA
1200
.
Q.
~
;:; 'E' o
800
'"
'0
,1L\ ,
\.
"
,
400
\ \.._~-
o o
5
10
20
15
25
30
Iteration teD
Fig. 6. 12
Change process of sum of object and penalty
The initial and optimal design parameters arc listed in Table 6. 6. Compared with the initial, the evaluation index of optimum is lower, but satisfied with the constraint of frequency difference. Restricted to the processing technique, the final design is obtained by retaining the first decimal place. Table 6. 6
Design variable
State variable
Initial and optimal design parameters of TRUM-40 stator Variable
Unit
Initial value
Optimal value
Final value
PI
mm
20. 0
17.635
17. 6
Pc
mm
0.5
0.506
0.5
P7
mm
0.5
0.694
0.7
ps
mm
0.5
0.501
0.5
pg
mm
2. 0
2.001
2.0
PlO
mm
0.5
O. 504
0.5
!::,f
kHz
0.987
2.096
2. 191
fo
kHz
37.010
10.356
10.321
Wov."
p'm ·m/s
6.639
5.276
5. 150
The finite element model of the final stator is established and solved to obtain and other modes shown in Fig. 6. 13.
B07
Chaptcr 6
Dcsign and Manufacturc of Traveling Wavc···
179
FEM model ofTRUM-40 talor
B07(4032 IHz)
Interference rnode(38 130Hz)
Interference rnode{4 760Hz)
B..(31 27 1Hz)
B",,(5 1 00 I Hz)
Fig. 6. 13
B07
modc and ncarcst modcs of stator for TRUM-10
According to the final design, the stator manufactured is tested with PSV300F-B, the result is shown in Fig. 6. 11. The three peaks in Fig. 6. 11 are corresponding to B06 • B07 , and B08 modes. respectively. Exciting at the fixed frequency 40 143Hz. the axial amplitude of TRUM-40 stator is 3fLm. 20
~ ~
"""2
"§,
"0
15
40 143Hz
10
'" ::E
5
0
J\..... 30
Fig. 6. 14
6.6
Frequency/kHz
40
50
Frequency responses of stator for TRUM-40
Adjustment of Two Phase Modal Frequencies of Stator
It is important for the good mechanical characteristics and stable operating of USM to make the frequencies of two orthogonal modes of the stator to be the same. In reality, due to the fabrication error and heterogeneous materials, the two frequencies do not coincide with each other. Fig. 6. 15 shows the modal nephogram and the frequency response characteristics of a TRUM-60 stator. When the driving voltage of phase A is applied to the stator. if the frequencies of
180
Ultrasonic Motors Technologies and Ap plicalions
the two modes coincide with each other, only one peak excited by A phase appears in the frequency response curve. corresponding to one modal shape; if their frequencies do not coincide with each other, peaks A and B excited by A phase appears in the curve. When the driving voltage of B phase is applied to it, the corresponding modal nephogram can be obtained. as shown in Fig. 6. 15(b). In this case the excitation frequency is elose to those of peaks A and B, then the response won't be satisfactory. Moreover, a rotary mode can also be excited by only one phase excitation. This is why a traveling wave can come into being in a stator with only one phase. Therefore. when the two phase excitations are made at the same frequency. the amplitudes of two standing waves arc different. According to Chap. 5. the traveling wave in the stator is distorted, which will result in the decrease of the performance and the unstable speed of the motor. This section presents a modification technique for the stator to make the modal frequencies of two phases coincide with each other. ~
2
E E
"=' 1.5
"
"0
.~
Q.
...
E
C
-g ~
0.5 0
34
(a) Mode for A phase
Fig. 6. 15
6.6.1
40
(b) Mode ror B pIJa e
Frequency response curve and modal nephogram of TRUM-60 stator
Method of Adjusting of Two Phase Modal Frequencies
The modification method for the stator is proposed based on structural perturbation theory. An appropriate mass or stiffness is added on the stator to modify its modal frequencies. A portion cut from the stator can be considered as the addition of the negative mass or stiffness. The structure modification is small, so it can be assumed that the modal shapes change little- 2 ]-. According to Thomson's Structural perturbation theory:22 23J , the perturbation mass or stiffness have an effect on kinetic energy and potential energy, respectively. Based on Lagrange equations, the dynamic equations for the modified stator can be written. The new modal frequency can be calculated by solving the dynamic equations. The modification method can be
Chaptcr 6
Dcsign and Manufacturc of Traveling Wavc···
181
obtained by the analysis of the computation result. Assuming V" P,' and C, are the volume, density, and stiffness matrix of a stator, respectively. The distributing mass with density PI and volume VI' whose deformation is ignored, is added to the stator. Meanwhile some material with stiffness matrix C2 and volume V, , whose mass is ignored, is placed on the stator. The derivation of Eq. (5. 28) in Chap. 5 leads to the stator vibration displacement, velocity and strain vectors as follows
CPmq
(6. 18a)
U = CPmq
(6. 18b)
e=cp'mq
(6.18c)
u
=
where CPm is the shape function matrix, CP:n is the strain matrix. Eq. (6. 18c) depicts the transformation between strain tensors and modal coordinates. Considering the added mass and stiffness, strain energy and kinetic energy of the stator can be written as
f
~
=
~ L/,uTu dV + ~ L/IUTU dV
v, eTC,
e dV +
~
f
=
V
2
eT C2 e dV
(6. 19a) (6. 19b)
Inserting equations above into Lagrange equation without dissipation leads to
1,2,···
(6.20)
where CPmk is the kth column of the matrix CPm. From Eq. (6.20), the modal frequencies of the stator before and after modification can be calculated as follows
(6.21)
[L,
cp'mkTC, cp'mk dV
f
Vs
+~
CP~'k P, CPmk dV + :i= .i
L,
f
cp'm/'C, cp'm} dV I t (6. 22) cp~,} Pi CPmj dV
V1
The analysis of Eq. (6. 22) leads to the following conelusions: (1) If only the ametabolic distributing mass with density PI and volume VI added to the stator, the modal frequency becomes
IS
(6.23)
182
Ultrasonic Motors Technologies and Ap plicalions
It is lower than the old modal frequency. (2) If some material with stiffness matrix C, and volume V, , whose mass is ig-
nored, is added to the stator, the new modal frequency is
(6.21)
It is higher than the old modal frequency. (3) If the hole is drilled somewhere with volume Vo in the stator, the modal frequency is
f
cr(k T C, cP'mk dV - ~
r LcP~'k v
f
1
cP'm, T C, cP'm; dV- ,-
v
~ ( cP~,; p, cPm;dV
p, cPmkdV -
(6.25)
According to weighted orthogonality of modal shape function, Eq. (6.25) can be induced into
Kk - K'k ----''--------';Mk -M'k where
Kk Mk .6
=
= =
W
iv, f
V,
cP:nk T C, cP:nk dV,
,
K'k
=
(6.26)
~w
;::::::::: Wk -
f cP:nk cP:nk dV f cP~k cPmk dV T
C,
(6. 27a)
Vo
cP~k p,
(K'k -
wi
Mk
cPmk dV , M'k)
M'k
=
p,
(6. 27b)
Vo
(6. 27c)
where Kk and Mk are the kth modal mass and modal stiffness of the original stator, respectively. K'k and M'k are the kth modal mass and modal stiffness after modifying the stator, respectively. Considering Eq. (6.26) , we can draw the conelusions as follows: If .6 w <0, the modified modal frequency is higher than the original one. If .6 w >0, the modified modal frequency is lower than the original one. Therefore, drilling hole at someplace can increase or decrease the modal frequency. The drilling hole is more feasible than the other ways. The following section will define the position of the hole.
6.6.2
Example of Adjusting of Two Phase Modal Frequencies
The finite element model of the modified stator is proposed using ANSYS. The K~ - wiM~ value of every element in the case of B09 mode can be calculated from the element table of A='JSYS software, which is shown in Fig. 6. 16. From Fig. 6. 16, it is shown that the (K:- w;M~) values of the tooth groove located at the wave crest or trough are big positive values. If the part of the
Chaptcr 6
Dcsign and Manufacturc of Traveling Wavc···
- . 1 e-04
Fig. 6. 16
- .7c- 04
- Ae - 05
Distribution of
- .Se - 06
183
.3c - OS
K:- wrM: for Bog mode of stator
tooth groove at the wave crest or trough is cut, the modal frequency will decrease. On the other hand, the (K'k - wiM'k) values of tooth top located at the wave crest or trough arc big negative values. If the part of tooth top is cut, the modal frequency will increase. At the nodal diameter, whether the part of the tooth groove or tooth top is cut, the modal frequency changes very little. USM's operating depends on two orthogonal modes. So one point located at the wave crest of the modal shape of a stator, must lie at the node radius of another one, as shown in Fig. 6. 15. In order to make the two modal frequencies the same, we shall cut the tooth groove, which lies at the wave crest or trough of the mode with higher frequency. In this way the higher modal frequency will decrease while the other modal frequency changes very little, so the two modal frequencies can be equal to each other. An application example is presented·"·. Fig. 6. 15 shows that the two modal frequencies measured arc 36. 81 kHz and 37. 23kHz, respectively. And the difference between the two modal frequencies is o. 42kHz. Using the PSV-300F-B we can find the operating mode shape with higher frequency, and deepen O.1mm on the tooth groove at the wave crest of the mode, as shown in Figs. 6. 17 and 6. 18. The result of the modified stator is obtained by PSV-300F-B. It is noticed that the two modal frequencies are 36. 67kHz and 36. 63kHz, respectively, and that the difference of them is only o. 04kHz now. The difference between two modal frequencies decreases by ten times. Recently, a new hole-drilling device, which can automatically adjust the two modal frequencies to coincide with each other, is fabricated by PDLab. It is applicable for the modal frequency modification of traveling wave type rotary USMs and rod shape rotary USMs.
6.7
Analysis of Flexible Rotor
In the early development of an ultrasonic motor, the disk spnng was used between the stator and rotor to produce the pre-pressure and the contact pressure between the stator and rotor, as shown in Fig. 6. 19. The rotor is called rigid ro-
Ultrasonic Motors Technologies and Ap plicalions
184
tor, whose deformation is considerable small and ignored.
6
~ <= <=
<:"" 'u
4
0
-.;
>
2
36
35
Modified stator
Fig. 6. 17
37
38 [ 1kHz
39
40
Fig. 6. 18 Frequency response of modified stator for TRUM using one phase excitation
for TRUM
The presented model in Chap. 5 is based on rigid rotor. Later we found that the contact interface between the stator and rotor is not real plane, and the radial speed of the point on the stator surface is large when the stator vibrates. So the power losses of the motor would be considerable. Then the rotor with appropriate fle.Libility is designed to improve the contact performance and reduce the power losses from the radial friction between the stator and rotor. As shown in Fig. 6. 20, the disk spring is deleted and the pre-pressure between the stator and rotor is obtained by the deformation of rotor. The new type of rotor is called flexible rotor in contrast to the old rigid one.
Shell
Washer Absorber Rotor
PZT
Fig. 6.19
TRUM with rigid rotor
Fig. 6. 20
TRUM with flexible rotor
In fact, if the excitation voltages arc applied to an ultrasonic motor with flexible rotor, the stator will vibrate as shown in Fig. 6. 21, while the rotor will also vibrate, as shown in Fig. 6. 22.
Chaptcr 6
Fig. 6. 21
Dcsign and Manufacturc of Traveling Wavc···
Modc shapc of stator
Fig. 6. 22
185
Similar modc shapc of rotor
The rigid-flexible coupling dynamic equation of the flexible rotor can be obtained by FEM['s:. The modal frequency, mass and stiffness are obtained by solving the eigenvalue equations. The dynamic equations of the motor are derived from the combination of the semi-analytical model of the stator in Chap. 5 and the rigid-flexible coupling dynamic equations of the rotor. Then the mechanical performance can be obtained by numerical analysis under a certain initial condition. The mechanical performance of TRUM-60, such as the rotary speed under a certain load, can be predicted by substituting the structure parameters, material parameters and the assembly relationship of the motor into the simulation model. The load characteristics can be simulated by calculating the rotate speed with different loads.
6.7. 1
Importance of Rotor's Flexibility for Performance of Motor
According to the analysis of the contact pressure between the stator and rotor in Section 5. 3, the rotor with an appropriate structure can not only simplify the structure of the motor by canceling the disk spring, but also can reduce the abrasion between the stator and rotor by causing the radial displacement of the contact point on the rotor. In the following section the contact force is analyzed with varying load, and the importance of the flexibility of the rotor to the motor is explained. At certain load, the steady vibration of the stator and rotor is investigated, and the radial speed and axial speed within one wavelength can be obtained as shown in Fig. 6. 23. The direction of the friction force IS determined by the relative speed of the point on the contact interface between the stator and rotor. The differences of radial and circumferential speeds of the contact point between the stator and rotor should be small and large respectively. In this case the smaller radial and larger circumferential components of the friction force will be obtained, and then the larger output torque can be possible. The traveling wave comes into being in the stator, which will excite a similar traveling wave in the rotor through the contact interface. The two traveling waves are not synchronous and there is a phase shift ([! because of the damping, as
186
Ultrasonic Motors Technologies and Ap plicalions
xl0-6
xlO- 4
4~~--F-n-c-tio-n------------:~A~m~0~u~n7t~ofrc~0~m~p~re~ss~io~n~0~f~fi~ric~t~io~n'la~y~er~:----,4 layer I--_...:.'P_----;:~ ,.- - _.- . 2 2 Stator - - - Rotor
o
"
o
" "
Axial displacement of n;;:o";: , - - Contact area
-2 Axial _ displacement stator _- L_ of _
_4L-_~
5
10
-L_~LL
15
20
25
..i -
-2
_ _~_ _L -_ _L-_~~_~_4
30
35
40
4~
50
0.4 ,----------::;;;..;...""""'1'=--------.------;----,0.04 I -:Stator j .... . ...... :Rotor .. 0.2 0.02
i:
!:
o
1 I
o
i:
-0.2
Jq
.............
10
15
20
I I
25
i:
!:
'<2 30
35
40
-0.02
45
Angle coordinate in the circumferential direction/CO)
Fig. 6. 23
Displacement and speed distribution
of points on stator and rotor surfaces
shown in Fig. 6. 23. In this case, the difference of the radial speed between the stator and rotor can be described as below. First, in the contact area denoted by A2 , the points at the contact interface between the stator and rotor have the radial speed with the same direction, which reduces the radial slip and the radial friction forces. However in the contact area denoted by Aj and A3 , the contact points of the stator and rotor have the radial speed in the reverse direction, which increases the radial friction and corresponding power losses at the interface. So the phase difference between the responses of the stator and rotor is set to keep the radial speed of stator and rotor in the same direction, in other words the spatial phase difference between the traveling waves of the stator and rotor is half wavelength, so that radial friction is reduced and the output efficiency of ultrasonic motors is improved. When the rotor vibrates, there exists a phase difference between the traveling waves in the stator and rotor because of damping. So the rotor can be designed by making the modal shapes and frequencies of stator and rotator almost the same. Moreover the vibration phases of the stator and rotor are opposite each other, which will decrease radial friction at the contact interface, and increase the torque and efficiency of the motor. If the modal frequency of the rotor is much lower than operating modal frequency (c. g. 38kHz), the phase difference between the responses of stator and rotor will reach half of the traveling wave wavelength. The radial speed of the points on the stator and rotor is in the same direction, but the radial speed of the points on rotor is small enough to reduce the radial friction between the stator and rotor and the radial component of the friction force at the contact interface is
Chapter 6
Design and Manufacture of Traveling Wave'"
187
still considerable. The phase difference will be small when the modal frequency of the rotor is elose to the operating frequency of the motor, which will decrease the radial component of the friction force at the contact interface. In this case, the contact area, where the radial speeds of points on the stator and rotor have the same direction, will decrease, but the radial speed of the points on the rotor will increase, which is advantageous in decreasing the radial friction component. The phase difference will come to zero when the modal frequency of rotor is much bigger than the operating frequency, which makes the radial speeds of the points on the stator and rotor opposite each other completely. In this case. though the radial speed of points on the rotor is small, the radial slip is very serious. So the rotor's optimal design is very necessary: at operating frequency, the mode shape of the rotor is similar to that of stator, such as the mode frequency of B09 of TRUM-60, elose to the operating frequency of the motor, and the larger radial speed response of the rotor can be obtained. Also, the modal frequency of the rotor should be a little lower than the operating frequency of the motor, and then the small phase difference can be obtained between the responses of the stator and rotor. Figure 6. 21 shows the simulated curve of the friction forces within one wavelength at different modal frequencies of the rotor when the operating frequency of the motor is 38kHz. It is found that the circumferential component of the friction force between the stator and rotor is larger when the modal frequency of the rotor is little less than the motor operating frequency. In this way the output torque of the ultrasonic motor increases. Figure 6. 25 shows the testing performance curves of the ultrasonic motor at the operating frequency 38kHz when the rotors with different modal frequencies are gIven, respectively. 0.Q2
~
~ 0
"-E 1> .~
'6
0.01 0
I. 32kH z,radi ai
-0.01
2. 385kHz,radiai
~ 3.375kHz,radial .£ -0.02
.2 t
'c
u.
4.32kH z,circumferentiaI4 -0.03 5. 38 .5kHz,circumferentia I6
- 004
6. 37 .5kHz,circum ferential 5
10
15
20
25
30
35
40
45
50
Angle coordi nate in the circu mfere nt ial direction/(Q)
Fig. 6. 24
Simulated friction force distribution in the contact
area at various modal frequencies of rotor
From the above analysis when the modal frequency of the rotor is a little lower than the motor operating frequency, the radial component of the friction force can be reduced and the output torque and efficiency of the ultrasonic motor will be improved. Generally the radial speed of the point on the stator is larger than
Ultrasonic Motors Technologies and Ap plicalions
188
200
<::::-''i ···<:·:....· ···X :.: :
..
~
~ ""~
150
40
I. 37 ,8kHz
1
2
3
.. .....
100
2, 38,3 kHz 3.33.6kHz 4, 37 ,8kHz 5, 38.3kHz 6.33.6kHz
30
'#-
~ 'u ""
20 Ii:
e.""
c
'" "0
10 0
~
0,2
0,8 Qutput torq ue I(N · m)
0.4
0.6
1.2
1.4
0
Fig. 6. 25 Output speed and efficiency vs. output torque for TRUM at various modal frequencies of rotor
that of the point on the rotor, so the radial speed of the point on the rotor should be as large as possible. That is why the inner part of the rotor is much thinner than the edge.
6. 7. 2
Comparison of Contact Area of Rigid and Flexible Rotor
The ultrasonic motor produccs torque by the elliptical movemcnt of the points on thc stator, which drives the rotor through the contact intcrfacc. So the morc points on the stator can propel the rotor at thc interface. the larger torque can bc obtained. That is to say, the contact arca of the stator and rotor is important to thc performance of the ultrasonic motor. Figure 5. 18 shows that the axial displaccment distribution along the radial coordinate of the points on the stator. Moreovcr thc axial displaccment increases with the larger radius. In order to obtain the contact area between the stator and rotor, the rotor should have a certain deformation to match the deformation of the stator. In this way the larger torque and rotary speed of the ultrasonic motor can be obtained. The rotor with large stiffness has little deformation and its bottom face keeps flat while the disk spring produces required pre-pressure. In this case the contact area is very small. only at the outer edge of the stator and rotor. as shown III Fig. 6. 26, which rcsults in thc small output torque of the ultrasonic motor.
7:::=:=:';,
:cs::::s::::s:'~'~~~::s;;::::¢::r- ~~~~~
)::::::::::::::::::::::::::::.........
Flexible rotor
Vibrating stator
Fig. 6. 26
Contact state between sta tor and rigid rotor
Vibrating stator
Fig. 6. 27
Contact state between stator and flexible rotor
Chapter 6
Design and Manufacture of Traveling Wave'"
189
The deformation exists in the flexible rotor due to the pre-pressure, which is specially designed. If the appropriate design for the flexible rotor is given. the deformation of the rotor will be the same to that of the stator. In this case the stator and rotor have an excellent contact with each other in the radial direction, as shown in Fig. 6. 27.
6.7.3
Effect of Rigid and Flexible Rotor on Mechanical Characteristics
The stiffness of rigid rotor
IS
so large that its vibration can't be excited. Then
the rotor has almost only rotary displacement around the shaft. According to Section 5. 3 the speed distribution of the stator and rotor at the contact interface has effect on the direction of friction forces, corresponding power losses, and even the output torque and efficiency of the ultrasonic motor. So the mechanical characteristics of the motor can be improved through the good design of the flexible rotor. The contact forces of the rigid rotor and flexible rotor in one wavelength are compared, as shown in Fig. 6. 28. It is found that the elastic deformation of the flexible rotor changes the distribution of friction forces. In this way the circumferential component of the friction force increases and the radial component decreases. In this case the stator can propel the rotor much better and the output torque and operating efficiency of the ultrasonic motor is improved. The mechanical characteristic curves of TRUM with the rigid and flexible rotor in Fig. 6. 29 validates that. 0.015 ,---------,------------;----=:-----------, I.C ircumferential direction fo r rigid rotor 2.Radial direction for rigid rotor 3.Circumferential direction for flexible rotor 1'Fi""'--~ 4.Rad ial direction for fl exible rotor
0.01 0.005 Ol----~
- 0.005 -0.01 -0.015 -0.02 -0.025 -0,03 -0.035
L------L-------'-_-'------L-------'-------'-".L-----L-------'-_-'------'
o
5
10
15 20
25
30
35 40
45 50
Angle coordinate in circumferential direction/(O)
Fig. 6. 28
Contact pressure distribution in contact
area for the rigid rotor and flexible rotors
6. 7. 4
Design and Manufacture of Flexible Stator
A rotor is an important part of an ultrasonic motor, which affects the transforming efficiency of the mechanical energy from the stator to rotor.
Ultrasonic Motors Technologies and Ap plicalions
190
150 .--------------------------,100
~.
~
~ Q) Q)
P.
speed { I. rigid rotor 2. flexible rotor
so
120 90
60
60
40
30
20
;fl
%
Elf.
{ 3. rigid rotor 4. flexible rotor
.&
ifJ
O~
0.00
iUl:E
__~L__ _~_ _ _ _~_ __L~~L-~O 0.2
0.4
0.6
O.S
1.0
Output torque/(N om)
Fig. 6. 29
Output characteristics of TRUM with the rigid and Ilexible rotor
1. Dynamic properties of rotor First, the radial deformation mateh of stator and rotor should be considered. Then the modal frequency of the rotor( the corresponding modal shape is similar to that of the operating mode of the stator) should be little lower than the operating frequency. Furthermore, the modal testing of a rotor is conducted. Fig. 6. 30 shows the measured frequency response curve of the rotor.
~
2
" ~
1.5
E
.",
.E 0.
""" 05
.",
<>
0-
<Jl
0 20
30
40
50
11kH z
Fig. 6. 30
Measured frequency response of rotor
2. Dynamic strength of rotor The inner web is very thin, so that the large deformation of flexible rotor can produce enough deformation and the radial slip between the stator and rotor decreases. In this case the rotor rotates and vibrates in the axial direction while the ultrasonic motor operates. Besides, the vibration amplitude of the rotor at the outer edge is very large, which makes the rotor fatigue damage, as shown in Fig. 6. 3l. So the dynamic strength must be taken into account while the rotor is designed. The damping material can be bonded to the rotor to reduce the vibration amplitude of the rotor and avoid fatigue damage. Fig. 6. 32 shows the measured response of the rotor using PSV-300F-B before and after the damping material is bonded. It is found that the biggest displacement amplitude of the rotor
Chaptcr 6
Dcsign and Manufacturc of Traveling Wavc···
191
without the damping material reaches 3fLm, while the rotor with the damping material only o. 08fLm. In this way the displacement response of the rotor at the thin edge is greatly reduced and the life of the rotor is prolonged.
(a) Wilhoul damping malerial
Fig. 6. 31 Damagcd flcxiblc rotor due to vibration
(b) Wilh damping material
Fig. 6. 32 Rotor with or without damping material
3. Selection of rotor material and friction material In order to decrease the rotor inertia and to improve the control characteristics, the rotor is usually made of duralumin. The friction material is applied to reduce abrasion at the contact interface and improve the life of an ultrasonic motor. According to the analysis in Section 6. 3, the friction force between the stator and rotor not only provides the driving force for the rotor but also brings some power losses, which will consume the mechanical energy of the stator and reduce the operating efficiency of the ultrasonic motor. So the friction coefficient of the material can not be too large. Although the material with large friction coefficient can produce output large torque, power losses at the contact interface is considerable and the efficiency is small.
6.8
Manufacturing Techniques of TRUM
The manufacturing process of TRUM with flexible rotor in Fig. 6. 20 is illustrated in Fig. 6. 33. As a novel kind of precise machinery, the attention should be paid to the following process:
1. Piezoelectric ceramic ring The silver electrode of fired piezoelectric ceramIc nng must be precisely divided based on the size requirements, and the piezoelectric constant d" should be uniformly distributed to each sector area, with the uniformity error no more than 2 to 3 percent.
2. Stator As the key part, the stator of the ultrasonic motor requires high manufacturing accuracy. The depth of stator teeth must be uniform, otherwise they will cause the difference of the two phase modal frequencies of the stator. The flatness of the bonding surface and good adhesion arc necessary to avoid the fracture of pie-
192
Ultrasonic Motors Technologies and Ap plicalions
Fig. 6. 33
Manufacture flow of TRUM
zoelectric ceramic ring in the operating process of the motor. Due to the small axial amplitude of the stator, usually only of micron level, the good flatness « 1. 5 p.m) and roughness « O. 5 p.m) are required for the ideal contact surface of the stator and rotor to obtain a certain output power.
3. Bonding There are two bonding parts in the assembly process of the ultrasonic motor: one is between stator and piezoelectric ceramic ring, the other is between friction material and rotor or stator. The bonding between the stator and the piezoelectric ceramic ring is very important. If the bonding adhesive layer is too thick, it will probably insulate piezoelectric ceramic ring and the stator substrate, though the bonding can be very strong. More mechanical energy will be dissipated when energy transfers from the piezoceramic ring to the metal stator substrate. When the same excitation voltages apply to the stator, a thick adhesive layer results in
Chaptcr 6
Dcsign and Manufacturc of Traveling Wavc'"
193
the low electric field applied to piezoceramic ring and bad heat conduction. If adhesive layer is too thin. it is difficult to ensure that the bonding between piezoceramic ring and metal substrate is strong enough. Usually. the thickness of the adhesive layer between piezoceramic ring and the stator substrate is about 3-5fLm. The same consideration is also given to the adhesive layer between the rotor and friction material. The thick adhesive layer will affect heat conduction and reduce the contact stiffness of friction layer. Consequently. the output performance of the ultrasonic motor decreases. On the contrary. the thin adhesive layer may affect the adhesive strength. Usually the thickness of the adhesive layer is about 5-10 fLm. Bin shi. Minqiang HU_26J. et al. presented a lot of experiments about bonding technology and corresponding experience.
4. Pre-pressure According to the dynamic simulation of TRUM in Chap. 5. the pre-pressure has a significant effect on the performance of TRUM. Ultrasonic motors with inappropriate pre-pressure can result in bad performance. serious heat. large noise as well as large input current. Smaller pre-pressure for the ultrasonic motor with good stability and silent operating results in larger speed. larger pre-pressure is necessary for the large torque of TRUM. Generally speaking. the pre-pressure of TRUM-60 is about 300N. TRUM-15 about 150:'\1 .and TRUM-30 about 60:'\1.
References [ 1
J
[ 2 [ 3
J J
[ 4
J
[ 5
J
Chunsheng Zhao. Development, application and future o[ ultrasonic motor. Electric Age, 2001(1): 1-3. (in Chinese) T Sashida, T Kenjo. An introduction to Ultrasonic Motors. Oxford: Clarendon Press, 1993. Bangehun Wen, Guozhong Zhang, Hongyi Liu. Comprehensive Design Theory and Method Toward General Quality. Beijing: Science Press, 2007. (in Chinese) Xiangdong Zhao, Bo Chen, Chunsheng Zhao. Characteristics estimation and optimal design o[ traveling-wave type ultrasonic motor. Small and Special Electrical Machines, 2003, 5: 1315. (in Chinese) Chunsheng Zhao. Research and development of ultrasonic motor in Nanjing University of Aeronautics and Astronautics. Journal of Vibration, Measurernent and Diagnusis, 2005, 25
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[ 8
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[9
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[10J
[llJ
(3): 167-173. (in Chinese) Xiangdong Zhao. Study on the Dynamic Modeling and Simulation of the Traveling Wave Type Ultrasonic Motor. Dissertation for the Degree of Doctor of Philosophy. :'-Ianjing: :'-Ianjing University of Aeronautics and Astronautics, 2000. (in Chinese) linbo Liu. Study on Mathematical Model and Drive System of Ultrasonic Motor. Dissertation [or the Degree o[ Doctor o[ Philosophy. Hangzhou: Zhejiang University, 1998. (in Chinese) Chao Chen. Study on Theoretical Model of Traveling Type Rotary Ultrasonic Motor. Dissertation for the Degree of Doctor of Philosophy. :'-Ianjing: :'-Ianjing University of Aeronautics and Astronautics, 2005. (in Chinese) Jianhua Rong, Jianlong Zheng, Feihong Xu. Dynamic Medication and Optimal Design. Beijing: China Communications Press, 2002. (in Chinese) Chunsheng Zhao. Some proposals [or development o[ ultrasonic motor techniques in China. MicromotorsServo Technique, 2006, 39(2): 61-67. (in Chinese) Hua[eng Li. Study on Driver of Ultrasonic Motor. Post-doctoral Report. Nanjing: Nanjing
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[l2J [13J
Huafcng Li, Chunsheng Zhao. Research on the ultrasonic motor driver based on LC resonant. Proceedings of the CSEE, 2005, 25(23): 111-118. (in Chinese) Hua[eng Li, Chunsheng Zhao. Micro-driver [or ultrasonic motor based on CPLD. Proceedings of the CSEE, 2005,25(7): 115-118.
[l1J
Xiangdong Zhao, Yikun Yuan, Chunsheng Zhao. Finite element analysis and modal mixture o[ ultrasonic motor stator. Journal of Vibration and Engineering, 2004,17 (S): 866-868.
[15J [l6J
Zhi[ang Fu, Hongxing Hua. Modal Analysis Theory and Application. Shanghai: Shanghai liaotong University Press, 2000. (in Chinese) linsong Zeng, Chao Chen, Chunsheng Zhao. An effective method of solving modal mixture of ultrasonic motor stators. Piezoelectric and Acuustooptics, 2006, 28(6): 722-724. (in Chinese)
[l7J
Shengqiang Zhou. Research on modeling, simulation and optimization of traveling ware type rotary ultrasonic motor. Post-doctoral Report. ='Janjing: Nanjing University of Aeronautics
[l8J [19J
and Astronautics, 2009. (in Chinese) Meiling Zhu. Study on Piezoelectric Traveling Wave Type Ultrasonic Motor. Post-doctoral Report. Nanjing: Nanjing University o[ Aeronautics and Astronautics, 1996. (in Chinese) Zhirong Li. II. Fundamental Study on the Cylinder-sphere USM with Multi-degree of Freedom and Its Control Techniques. Dissertation for the Degree of Doctor of Philosophy. :'-I an-
[20J
jing: :'-Ianjing University of Aeronautics and Astronautics, 2001. (in Chinese) Chao Chen, ]insong Zeng, Chunsheng Zhao, et al. Parametric optimization o[ stator o[ traveling wave type ultrasonic motor. China Mechanical Engineering, 2009, 20 (5): 568-572.
[21J
R V Grandhi. Structure optimization with frequency eonstraints-a review. AlAA, 1993, 31 (12): 2296-2303.
[22J
Wenliang Wang, Zuorui Du. Structure Vibration and Dynamic Substructure Method. Shanghai: Fudan University Press, 1985. (in Chinese)
[23J
W T Thomson. Theory of Vibration with Application. :'-lew York: Prentice-Hall, Inc. ,Englewood Cliffs. :'-I, 1981. ]insong Zeng, Chao Chen, Chunsheng Zhao. Adjusting traveling wave type ultrasonic motor
[24J
stator modal frequencies into coincidence. Journal of Nanjing University of Aeronautics & Astronautics, 2006, 5(38): 605-606. (in Chinese)
[25J
Shiji Sun, Chengxu Huang. Rigid-flexible Coupling Dynamic Analysis and Simulation of Mechanical System. Beijing: China Communications Press, 2000. (in Chinese)
[26J
Bin Shi, Minqiang Hu, Zhuangrui Zhu. Effect of adhesive on the vibration of the stator in ultrasonic motors. Proceedings of the CSEE, 2001, 21 (7): 72-77. (in Chinese)
Chapter 7
Bar-type Traveling Wave Rotary Ultrasonic Motors The bar-type traveling wave rotary ultrasonic motor (BTRUM) described in this chapter ineludes two types: single degree of freedom (SDOF) and multi degrees of freedom(MDOF) ultrasonic motors. Both of them use the vibration of the bartype round stator to achieve conversion from electrical energy to mechanical energy. This kind of structure makes the BTRUM possess a number of unique features, such as simplicity, cheapness, ease of processing, suitability for minitarization, etc. Therefore, it promises a vast potential market in micro air vehieles (MAV) , micro-robotics, precision instruments, medical equipment, and other industrial areas. Using the BTRUM of SDOF and MDOF developed by PDLab as examples, this chapter will illustrate their motion mechanism, design principles, optimization of the structural dynamics, drive and control techniques, and so on.
7. 1
Review of Bar-type Ultrasonic Motor
The BTRUM of SDOF uses the bending modes of the bar-type round (or square) stator to operate. From the review of certain research references, the motor mainly has the following three types: 1: .
1. Using metal pipe or metal cylinder as the stator Piezoelectric ceramic pieces or thin films are fixed to the metal body in different ways. In 1998, Morita used the hydrothermal method to coat a piezoelectric thin film on the metal surface and succeeded in fabricating an ultrasonic micromotor with 2. 4mm in diameterL2J. Furthermore, in 2000, the another same type of micromotor with 1. 1mm in diameter and 5mm in length was successfully fabricated by Morita L3J . In 2002, Koc developed a single phase micromotor using the bending modes of a hollow metal cylinder:4:. 2. Using P ZT piezoelectric ceramic cylinder or tube as the stator In 1989, Shimizu and Yoshida firstly used a piezoelectric ceramic rod to construct a vibrator based on two orthogonal bending modes L5J . In 2000, Shuxiang Dong used a PZT tube poled in four directions as a stator, and developed a BTRUM micromotor which is 1. 5mm in diameter and 7mm in lcngth[6:. In 2001, Tieying Zhou developed a micro BTRUM with 1mm in diameter and 5mm in length, and
C. Zhao, Ultrasonic Motors © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011
196
Ultrasonic Motors Technologies and Ap plicalions
it has been used to test in medical endoscope systems- 7 - 8 -
3. Using the Langevin vibrator as the stator In 1988, Kurosawa firstly developed a new BTRUM[9: , which possesses the motor with the similar structure studied by Tsinghua University-lo-llJ. Since 1997, sevcral kcy issucs of thc BTRUM, such as thc driving mechanism, optimal dcsign, dcvelopmcnt of new prototype motors, havc bccn systcmatically studied in PDLab. For cxamplc, the elliptical motion equation of thc driving point was firstly derived by Xiangdong Zhao, Changqing Liu, and Chunsheng Zhao 1l2 -. Thc conccpt of an cffectivc clliptical motion was firstly discusscd by Yinghui Dong and Chunshcng Zhao[13:. A ncw series of the BTRUM of SDOF werc proposed by Chunshcng Zhao, H ua Zhu, and Xianglin Ma, as shown in Fig. 1. 11 (b):14 17:. Thc BTRUM of MDOF is dcvelopcd on the basis of thc BTRUM of SDOF. Because this motor can provide the motion with two or more degrees of freedom, it promises a bright application prospect in micro robot joint, endoscope, scanning devices in microcamcra, and othcr fields. Thc idca of thc BTRUM of MDOF was first put forward by Amino: 18 ]. This motor operates with a first longitudinal mode and two third bending modes of a cylindrical stator. Three degree rotations of a rotor can be realized by combining any two of thesc thrce modcs. The stator is 20mm in diamctcr, 118mm III lcngth, and thc rotor is 25mm in diamctcr. Its maximum output speed is 100r/min and maximum output torque is O. 035N o m. Subsequently, Takemura developed the BTRUM with 3 degrees of freedom (lOmm in diamctcr and 30. 67mm in lcngth), and also invcstigated thc design mcthod, output performancc, control features, performance simulation, etc: 19 22:. Its stator used a first longitudinal and two second bending modes. Its rotor is 10mm in diameter, its maximum speed reaches 250r/min, and stall torque 7mN o m. In thc parallel with Takemura's work, Chunsheng Zhao and Zhirong Li also conductcd the rcsearch on the BTRUM of MDOF. The motor uscs thc first longitudinal mode and the second bending modesL23-33-. According to the number of the stator, the motor with MDOF can be divided into two types: single stator and multi-stator. Thc BTRUM of MDOF proposed by PDLab belongs to the typc of singlc stator. Rcaders can find morc investigations on thc motors of MDOF with multi-stator in Refs. [31J and [32].
70 2 70 20 1
Construction and Motion Mechanism of SDOF Motor Construction
The structurc of thc BTRUM of SDOF developcd by PDLab is shown in Fig. 7. 1 [14:. The motor is compos cd of a shaft 1, retainer ring 2, rolling bearing 3, lower mass 1, insulator piece 5, piezoelectric ceramic rings 6, friction piece 7, upper mass 8, sliding bearing 9, spring sets 10, spring 11, rotor 12, and pin
Chapter 7
Bar-type Traveling Wave Rotary Ultrasonic Motors
197
13. The piezoelectric ceramic rings are clamped by the upper mass and the lower mass through screws.
~}+ ~
PZTrm phase A
~
~} pZTfor ~
Elcclrode for ~ ground ignal
o
Electrode for vollage signal ............
Fig. 7.1
Structure of BTRUM
Fig. 7. 2
phaseB
PZT for _
- - Feedback phase
Layout of PZT
The configuration of the piezoelectric ceramic rings is shown in Fig. 7. 2. The feedback phase is used to detect the vibration of the stator, and to control the speed of the motor. When a sinusoidal alternating voltage is applied to the piezoelectric ceramic rings of phase A (the excitation frequency is close to the resonance frequency of the first bending modes). the stator will vibrate in the left and right directions. When phase B is imposed by a cosine voltage with the same frequency and amplitude, the stator will vibrate in the front and behind directions. These two perpendicular bending modes will be excited when the excitation voltages arc applied to phases A and B, respectively, and these two modes arc composed into a rotating bending mode (one traveling wave). At this time, the track of points on the driving surface will be an ellipse. Because of this elliptical motion and the pre-pressure between the stator and rotor, a friction force will be produced. The rotor can be driven in the direction that is opposite to the direction of the traveling wave. In order to clarify the driving mechanism. the elliptical trajectory equation of the driving points will be derived in the following section.
7.2.2
Motion Mechanism 1l2 ,3 1-
1. Traveling wave on driving sur face Assuming that the first bending mode frequency of the bar-type round stator is
pez;.
WI , 0 is the intersection of the axis of the stator and the end surface. y) is one of the driving points on the end surface of the stator, as shown in Fig. 7. 3 (a) , when the stator is independently excited by the voltage cOswlt of the phase A, and the influence of the modal damping and other vibration modes arc ig-
198
Ultrasonic Motors Technologies and Ap plicalions
nored. the displacement of point 0 in the
:x;
direction is (7.1)
and the rotation angle equation is (7. 2)
where U o and (30 are the amplitude of the horizontal displacement and rotation angle of point 0, respectively.
y
"p
(a) Location of poim P on end surface of stator
Fig. 7. 3
(b) Displacement of point P
Deformation of cnd surface of stator excited by phase A
Because the bending deformation of the stator is very small compared with its geometrical dimension and the deformation is consistent with the plane cross section assumption, as shown in Fig. 7. 3(b). the displacement of point P in the x and z directions can be expressed, respectively as (7. 3) w~ = -
rcose sin(3 ::::::::- r (30 cose cosw1t
(7. 1)
Similarly. when the stator is excited independently by the voltage sinw1t of the phase B, the displacements of point P in y direction and z direction can be expressed as Vp =
U o sinwjt
w~ = -
r (30 sine sinwjt
(7. 5) (7. 6)
If both phases A and B are used for exciting the stator at the same time, the displacement of point P in the z direction is (7.7)
From Eq. (7.7) , it can be coneluded that the displacement response of point P in z direction is decided not only by the time t. but also by the location angle e. This wave rotates along the circumferential direction. This proves that the BTRUM based on the bending modes belongs to the traveling wave type motors. When the displacement of point P is decomposed in the radial and tangential directions, as shown in Fig. 7. 4. the displacements in radial and tangential directions can be obtained, respectively:
Chapter 7
Bar-type Traveling Wave Rotary Ultrasonic Motors
199
u,
=
upcos6l+vpsin6l= UOcos(wlt-6I)
(7. 8)
u,
=
vpcos6l- upsin61
(7. 9)
=
Uosin(wjt- 61)
From Eqs. (7.8) and (7. 9), it can be found that the displacements of point P in the radial and tangential directions all can be expressed by the traveling wave equations.
2. Motion trajectory of driving point In order to facilitate the graphical expression. the trajectory of point P in the fourth quadrant of the xOy coordinate system is analyzed here. as shown in Fig. 7. 4(a). Dashed line represents the balance position of the stator, and solid line represents the position of the stator at a moment time t. z
x
Direction of the
traveling wave
Ca)
Fig. 7. 4
Cb)
Motion trajectory of point P in 3D coordinate system
Defining the tangent to the circle denoted with the dashed line through point P as axis Xl • and Po is the position of point P at the moment t when the traveling wave has been formed. line PoP ~ that is perpendicular to axis Xl is added as well as line Po P; that is perpendicular to the driving surface at the balanced position. So it can be obtained that PoP~
=
I Wp I = rf3ocos(wlt+ 61)
(7.10)
From Fig. 7. 1 (b) there are (7.11)
(7.12) From Eqs. (7.11) and (7.12), it can be found that the displacement amplitude of point P in the :rOy plane is a constant, and phase angle a is a function of time, namely (7.13) P~Peos(a
If the angle between line Po P
+
61)
=
UOeos(wlt+ 61)
(7.14)
and the driving surface at the balanced position
Ultrasonic Motors Technologies and Ap plicalions
200
can be defined as y , it yields that tany
=
PoP~ _ rflo PoP ~ Wo
(7.15)
-,-- -
From Eq. (7.15), it can be found thatyis a constant, which means that point P will move in a certain plane. In which we define the line that is parallel to PoP ~ and cross point P as axial Y l , the coordinate value of point Po in the X l PYI coordinatc system can be expressed as .LIP,
=
PP~
YIP,
=
PoP~
= =
-PP~sin(wlt+(J)
=
-Uosin(wlt+B)
J(rflo)'+U~COS(Wlt+B)
(7.16) (7.17)
So it can be obtained: 2
Xli'
U;'
+
,
=
YIP,
(rflo)'+U;
1
(7.18)
From Eq. (7. 18), it can be found that when the stator is excited by phases A and B at the same time, the motion track of point P is an ellipse. The tangential velocity of point P can be written as (7.19) 3. Effective elliptical motion According to Eqs. (7.7)-(7.9), if the displacements of the points in the radial, tangential, and z directions arc composed, the trajectory of point P can be obtained in various coordinate systems. (1) Composing the displacements of point P in tangential and z directions, it can be obtained: ,
2
~+~=1 (rflo)2 U;
(7.20)
Equation(7.20) shows that the projection of the trajectory of point P on the vertical plane is an ellipse, as shown in Fig. 7. 5. If the rotor is pressed on the end sur face of the stator, this elliptical motion will produce the torque to drive the rotor and it is the necessary condition to achieve the power transmission from the micro vibration to the macro rotation of the rotor. We refer to the elliptical motion as effective one. (2) Composing the displacements of point P in radial and z directions, it can be obtained that _
Wp - -
r flo U,u
r
(7.21)
Equation(7.21) shows that the projection of the trajectory of point P on the plane formed by the radial axis and z axis is a straight line, as shown in Fig. 7. 5. Addi tionally, the slope of the line is equal to - r flo IU o. If the rotor is rigid, this radial motion will yield the relative slip on the contact surface between the stator
Chapter 7
Bar-type Traveling Wave Rotary Ultrasonic Motors
201
Driving s urface . "' - ...
Elliptical motion tnljectory Effect ive el li ptical Illotion
-
x
o '"
Fig. 7. 5
Motion decomposition of point P
and rotor and result in dissipation of energy, thereby decrease the efficiency of the motor. In accordance with the above analysis, the motion of 16 points at four moments in one period is simulated and the result is shown in Fig. 7. 6. The three conclusions can be obtained: CDthe trajectory of points is an ellipse, and angle y exists between the elliptical plane and xOy plane; CZ)only when the displacement of point P reaches the maximum amplitude will it drive the rotor to rotate. So in one period, every particle will contact the rotor only once, and at every moment. only one particle will contact the rotor; Gil the moving direction of the traveling wave is opposite to that of tangential vclocity of the driving particles. Moving direction or the traveling wave
X
I
o
10--6
.........
.. .... -
Ell iptical motion direction or the driving particle
................. , J'f ............. !.. .
... ~ ..
-I - 0.02 x/m
Fig. 7. 6
7.3 7.3. 1
0.02
-0.02
Y/m
Motion trajectories of 16 points at four moments in one period
Optimal Design for SDOF Motor Design Principle
There are certain requirements for the design of the stator of BTRUM. First of all. the stator must have a proper operating modal frequency. A too low modal
Ultrasonic Motors Technologies and Ap plicalions
202
frequency will lead to the noise, lower vibration velocity and limit output power of the motor. If the modal frequency is too high, energy loss will increase inside the motor. So the efficiency of the motor will reduce. Experiments show that the proper modal frequency of the motor should be in the range from 30kHz to 10kHz. Secondly, there is a larger difference between the operating mode frequency and other mode frequencies for avoiding the modal interferences. Fig. 7. 7 shows the velocity response versus the frequency of the stator of the BTRUM. The bending mode frequency is 30. 2kHz. It can be found that one interference mode exists with the frequency 31. 8kHz. So not only the bending mode but also the ncar interference mode, which refers to a radial extension-contraction mode, will be excited when an alternating voltage is applied to the piezoelectric ceramic rings. The interference mode will yield more energy consumption and lower driving efficiency due to its influence on the contact condition between the stator and rotor. Generally, the difference between the operating mode frequency and the nearest interference modal frequency is preferably more than 2kHz.
~E
rI
800
'ri 600 "" .~ C.
~
Operalion mode Inlerference mode
400
E:-
-G 0
~
200 30
40
50
60
Frequency/k Hz
Fig. 7. 7
Velocity response versus frequency of BTRUM stator
There are also some requirements for the mode shape when designing BTRUM. Because the structure of the stator is asymmetric, using only one end surface for driving the rotor, the lower mass is heavier mainly to reduce the amplitude of the bottom and decrease energy transmission from the stator to the housing. Grooving on the upper mass is to reduce local stiffness, increase the amplitude of the end surface and regulate the modal frequency of the stator because the bending mode frequency is very sensitive to the depth of its groove. Fig. 7. 8 shows the relationship between the first bending mode frequency and the depth of the groove on the stator with diameter 20mm. In addition, the conversion from the electrical energy to mechanical energy of the ultrasonic motor is realized through the deformation of piezoelectric ceramic elements, which should be placed as elose as possible to the position, at which the strain of the stator is the largest to obtain higher conversion efficieney c05 06:. The modes of the stator depend on its structure, geometrical dimension and material parameters. For such a structure, "trial-and-error" method can not guarantee the design efficiency and can not satisfy the design requirements for
Chapter 7
Bar-type Traveling Wave Rotary Ultrasonic Motors
203
34
N
:r:
II
.:.<
~
30
ff
26
cJ)
22
ij
"""E0 "
-. ~-
'ti
ij
18 2,2
a:l
2,6
3.4
3
3.8
Depth of groove/null
Fig. 7. 8
Effect of trough on bending mode frequency
modes. Therefore, we must usc certain methods to solve the problem through multi-objective optimization. Considering that the dynamic model of the stator is the basis of structural optimization, the following discussion will focus on the theoretical model of the BTRUM stator.
7. 3. 2
Dynamic Model
Considering that the analytical method or semi-analytic method can only be used to solve the specific mode of relatively simple stator, and unable to meet all of the design requirements, so that in this chapter finite element methode FEM) is used to analyse the dynamic characteristics of the stator. Fig. 7. 9 shows the basic structure and structural parameters of BTRUM stator with SDOF. The stator is divided into 264 hexahedron elements with 408 nodes and 1 224 degrees of freedom. The vector, composed of the displacements of the nodes from :'\10. 1 to No. 11 in y direction, is used as the mode shape of the bending vibration. lZ !
I
2 :ll E
'"
3
~
4 5
Q.
0.
:::>
6 7
f-
N
0..
8 h,
Fig. 7. 9
9
10
Structure and FE model of stator
The displacement field of the element j can be expressed by the node displacements as ~~Xl
IVs l 3x3 ] ~~n
=
Nrn ~~n
(7.22)
204
Ultrasonic Motors Technologies and Ap plicalions
where l3x3 is the unit matrix, &.n = [UI VI WI Us Vs Ws T is the displacement vector of the element nodes, and N m is the matrix of the shape functions. Thc strain tcnsor E of thc elcment can bc cxpressed as (7.23) where Lm is the matrix of differentiation operators. By substituting Eq. (7.22) to Eq. (7.23), it can bc obtained that (7.21) To nodes of piczoelectric elemcnts, the electric potcntial vcctor has to be added. Because in practical application the piezoelectric ceramic ring is very thin and thc voltages are applicd to its surface, the electric potential function can bc approximated as (7.25) wherc qu and qd arc the potentials on the two surfaces of the ccramic ring. The electric field is simplified as a constant and only imposed to the ceramic ring in thc thickness direction. Thcreforc it can be exprcsscd as
o E
=
[E,
(7.26)
o
wherc hI' is the thickness of thc ring. The stress tensor ( j of a metallic element can be written as (j
=
(7.27)
C,E
where C, represents the stress stiffness coefficient matrix of the metallic element. If defining the poling direction of thc ring as z direction, from thc sccond piezoelectric cquation, the strcss tensor of the piezoelcctric elemcnt can bc written as (7.28) where C; and e are the stress stiffness coefficient matrix and the piezoelectric constant matrix of the piczoelectric matcrial, respcctively. Whether it is thc metal or the piczoelectric matcrial, thc kinetic encrgy of thc element j can be defined as Tj
=
~f ~JT ~JdV 2 vP
=
~ 2 ~JTM ~J m
mm
m
(7.29)
J
where p is its density and (7.30) For the metallic elcment, the potcntial energy can bc defined as
Chapter 7
Bar-type Traveling Wave Rotary Ultrasonic Motors
2If v
V~
T
e udV
2If v
=
T
e C, edV
=
IT
..
2~;" K:':m~;"
205
(7.31)
e
J
where (7.32) For the piezoelectric elements, the potential energy is expressed as (7.33) Substituting Eqs. (7.21) and (7. 26)into Eq. (7.33), it can be obtained that
V Jp -where
",jT KJmmUm "'J
21
Urn
jK:m ~ f JI'
21
-
BT CEB
-
m
"'JT KJme qe
Urn
dV
m
f Bm e B,dV Vj
K;", -
1
TI'
(7.34)
(7.35)
Vj
From the second piezoelectric equation, the electric potential energy of piezoelectric element j can be expressed as
Wi,
=
~
LET (ee +tE)dV
~ q~K~m~;n + ~ q~K;,q,
=
(7.36)
J
where
{ K~m K~,
=
=
K;",T
=
LB~eBmdV j
LB~t B,dV
(7.37)
J
where t is the dielectric constant matrix of the piezoelectric material. According to the Hamilton's principle, the energy equation for the stator divided into N elements can be represented as (7.38) where 1'1 , V j
,
and W j are the kinetic energy, potential energy, and electric en-
ergy of element j, respectively. oWr and oW" arc the virtual mechanical and electrical works done by the external force and the charge, respectively. Making the modal analysis of the stator, Wf=W,,=O. Substituting Eqs. (7.29), (7.31), (7.33), and (7.31) into Eq. (7.38) and performing variation of ~m and qc' the dynamical differential equation of the system can be obtained:
{
Mmm ~m K~,~m
+ Kmm
- K"q,
~m =
=
Qq
F, - Kmcqc
(7.39) (7.10)
where Mmm , Kmm' K m" and K" arc the mass, mechanical stiffness, piezoelectric
206
Ultrasonic Motors Technologies and Ap plicalions
stiffness and dielectric stiffness matrices of the stator, respectively. ()m is the displacement vector of the element nodes, (Fe - Km,q,) is the generalized force vector applied to the stator and F,. will be discussed in Section 7. 4. 2. Qq is the electric charge vector applied to the ceramic surface. The natural vibration equation of the stator can be expressed as
(7.41) Solving this generalized eigenvalue problem can lead to n eigenvalues and the corresponding eigenvectors, i. e. the modal frequencies and mode shapes. Table 7. 1 shows the comparison of the results calculated by two different methods with the experimental result when the stator material is stainless steel (3Cr13), and the piezoelectric material is PZT8. The calculation is conducted by this dynamic model (Matlab environment) and ANSYS software. It can be seen that these three results have reached good agreement and this dynamic model can be used to do the sensitivity analysis and optimization of the stator. Table 7. 1
Comparison of results calculated with experimental one
Modal frequency of the first bending vibration/Hz Tolerance relative to experimental result/ %
Calculated result by the author
Calculated result by ANSYS software
Experimental result
34 537
33 701
33 250
3.87
1. 36
Because the motor always operates elose to resonance, without other interference modes, ()m is simplified as
cp, ] [ql (t) ]
(7.42)
q, (t)
where CPl and cP, are the two orthogonal bending mode shapes, and q = [ql q2 ] T is the modal coordinate column matrix. Inserting Eq. (7.12) into Eq. (7.39), then pre-multiplied by cpT at both sides of the equation, it can be obtained that (7.13) where Fq = cpT F, is the modal force column matrix corresponding to the contact force vector F,. under the modal coordinate column matrix q, and K= - CPTK"" is the force coefficient matrix which represents the conversion capacity of the piezoelectric material from the electrical energy input to the mechanical energy. If the eigenvectors cA and cP, are normalized by the modal mass, then the modal mass, modal stiffness, and force coefficient matrix can be expressed as
(7.44)
o
Kmm
]
[W' 0
0 ] W'
(7.45)
Chapter 7
Bar-type Traveling Wave Rotary Ultrasonic Motors
207
(7.16)
7.3.3
Sensitivity Analysis
Because the structure is divided into many clements in FEM. the shape function is defined to calculate the mass and stiffness matrices of the element. In the global coordinate system. these elemental matrices are composed into a global matrix. which will form a vibration equation that is solved to get the modal parameter of the structure. Therefore. in the iterative optimization process. the mode required has to be identified from all eigenvalues and eigenvectors calculated by Eq. (7.41). MAC (Mode assurance criteria). which is used to identify the bending mode in this chapter. was originally applied to identify the difference between the theoretical and experimental modes in the modal analysis and testing. It can be expressed as
(MAC)i
l
(I/>~ 1/>,') (cfi[1 1/>01) •
=
1,2.3.··· .16
(7.47)
where I/>'i is the column matrix of the modal shape calculated by FEM. 1/>01 is the column matrix of the normalized reference mode. which is the first bending mode of the stator. and the latter can be expressed as follows:
1/>01
[4. 29
=
2. 92
- o. 96
- 1. 0
-
1. 13
-
o. 72
-
o. 37 o. 54
-
o. 56
o.
0.02
50
r
Here. taking i = 16. the first 16 modal frequencies and the corresponding MAC values can be calculated. The first 6 modes are rigid motion modes and the corresponding frequency values are equal to zero. So Table 7. 2 only lists 7-16 order modes. The eighth mode with the largest MAC value is the first bending mode. while the seventh mode with the almost same MAC value as that of the eighth mode is the orthogonal mode of the eighth mode. Mode calculation and corresponding MAC value MAC value Order flHz MAC value
Table 7. 2
Order
flHz
7 8
31 31 31 50 50
10 11
o. o. o. o.
537 538 656 499 499
996 998 670
12
022 O. 044
15 16
58 59 59 62 68
13
11
0.611 o. 177 0.251
898 553 551 554 238
o. 884 o. 888
As shown m Fig. 7. 9. the relation between the numbers of physical parameters and dimensions can be shown in Table 7. 3. Parameters PI • P, • P,. P5' and P7 are selected as the optimal variables based on the sensitivity analysis method in Section 6. 3. 2. Table 7. 3 Parameters and numbers
Dimension of the structural parameters
Structural design parameters of the stator
PI
pz
P,
P,
P5
P6
P7
rz
r3
r6
hi
hz
h3
hs
208
7.3.4
Ultrasonic Motors Technologies and Ap plicalions
Objective Function
The design requirements for the vibration mode of the BTRUM have been discussed in Section 7. 3. 1. Doing the optimal design, these requirements must be considered in the objective function F for the optimization algorithm, which ineludes: (1) The first bending modal frequency fbi and the target design frequency ft should be as elose as possible, that is FI
=
I
fbi - ft
(7.18)
I
(2) The amplitude of points on the driving surface should be as large as possible, while the amplitude of points on the lower mass should be as small as possible. If ¢tr and ¢tt x represent the first bending mode shape values in the x direction of Node 1 and :'\Jode 11 in Fig. 7. 9, respectively, this means
F
2
-I ¢i,i" 1
(7.49)
¢i,~
-
(3) Piezoelectric ceramics group should be placed on the antinode( with maximum strain) of the first bending mode of the stator, which means the position of Node 7 should be ncar the middle of the antinode, that is
(7.50) where ¢;,~ and ¢t~ represent the bending mode shape values in the 01': direction of Nodes 6 and 8 in Fig. 7. 9, respectively. (4) The difference between the bending mode frequency and the nearest interference one should be as large as possible, that is
F,
=
I
fO
1
(7.51)
fO
_ ill -
_ iut
Based on the above four requirements and taking into account differences of orders of magnitude, the partial objective functions arc multiplied by weighted coeffieients, and the global objective function for the stator design can be expressed as
(7.52) i=l
where Pi stands for the weighted coefficients. The mathematical model of the optimal design of the stator can be expressed as a minimum problem with boundary constraints:
,
min F
=
b PiFi
(7.53)
i-"j
j
=
1,2,1,5,7
where p;b and P't' represent the lower and upper boundaries of the optimal variables, respectively.
7.3.5
Optimal Algorithm and Results
The pattern search algorithm in Matlab toolbox is used in the optimization mod-
Chapter 7
Bar-type Traveling Wave Rotary Ultrasonic Motors
209
el L37 .• and the boundary of the design variables is defined as plb
=
[7
pub
=
[9. 5
1. 5
1
6. 5
6 1
1J 10
(7.54)
5J
The start values of the pattern search algorithm are p"
=
[7.75
6
2.7
8
(7.55)
3J
The vector in Eqs. (7.54) and (7.55) represents the dimensions of the design variables PI • P, • P4 , Pc , and P7 in mm. The stator modal frequency at the start point is 31 538Hz. the design goal frequency is 36 OOOHz and the weighted coefficients arc f31 = 1. f32 = 1. 5 X 10' • f33 = 50, and f3, = 1. 2 X 10 5. Figure 7. 10 shows the iterative process and the optimal results of the design variables. The values of the partial objective functions arc compared in Table 7. 4. where F represents frequency of first bending mode, A represents amplitude ratio of point on the driving surface to the one on the bottom of stator, D represents distance between Node 7 and the middle of the antinode of the bending mode, and T represents tolerance between the frequency of operating mode and interference mode. It can be the seen that after the optimization, the modal frequency of the stator is 36 003Hz, the piezoelectric ceramic group is closer to the center of the antinode of the bending mode, and the difference between the frequencies of the operating mode and the interference mode becomes larger. In addition, the amplitude ratio of the point on the driving surface to the one on the bottom of the s ta tor increases to 10. 79, as shown in Fig. 7. 11.
-§
7000
<E
5000
"
1000
g
Convergence of the objective function value is at I 486
:'" "5 3000
:0-
0
0
10
5
I5
20
25
30
35
40
45
50
Iterative times
~
0 .:;;
"E"
:0 -;;;
.;
C.
0
0.0 10 0.008 0.006 0.004 0.004 0.002
0.0077
0'1i 5 0.001 02 2
Fig. 7.10
Table 7. 4
0.0092
0.0065
n
5
3 4 De ign variables
Iterative process and optimal results
Values of the partial objective functions after the optimal design
F/Hz
A
D
T/Hz
Start values
34 538
7.65
40. 129
117.5
Optimal results
36 003
10.79
O. 101
1 522. 7
This numerical example shows that the optimal model can meet all the design
210
Ultrasonic Motors Technologies and Ap plicalions
7
After optimal design
4
"
~ ~
"
:;:;:
•
Before optimal dcsi!:,'ll
5
"d 0
Model proposed
Ansys
6
by author
Fitting curve
*
---
D
_ ...
0
_---
.. ~
2
D· •••.••. ". 0
2
I 2
4
'.
~-~~.-.-.-.-.-.-.-.-.-.-.-.
5
6
7
8
9
10
11
Numbers of nodes
Fig. 7. 11
Comparison of mode shapes bdore and after optimal design
requirements for the stator and it provides the theoretical foundation for the optimal design of the BTRUM with SDOF.
7.3.6
Modal Frequency Modification of Stator
The stator of BTRUM is axially symmetrical. and has two orthogonal operating modes with the same shape and same frequency. However. in reality. the two operating frequencies usually don't coincide with each other because of the influence of heterogeneous materials. irregular elamping pressure of screws or a machining error. Thus. when the stator arc driven by the two phase voltages at the same frequency. the amplitudes generated are different. Then the elliptical motion of the points on stator is distorted. resulting in the unstable speed of the motor. The BTRUM presented in this section is composed of one stator and two rotors. The characteristics of the velocity response versus frequency for the stator is shown in Fig. 7. 12. Solid and dash-dot lines indicate frequency responses excited by phases A and B. respectively. The mode shapes of two orthogonal first bending modes arc shown in two small figures. Their two frequencies arc 26. 56kHz and 26. 73kHz. respectively. The difference between the two frequencies is 170 Hz. which doesn't satisfy the requirements of the design. Generally the difference should be no more than 100Hz. To solve this practical problem. a modification technique is proposed which can adjust the difference to no more than 100Hz in an assembled state. 1. Finite element analysis In Section 6. 6. according to the structural perturbation theory. forming a recess portion in a predetermined portion of the stator. or adding an appropriate mass can change the modal frequencies of the stator. From Eq. (6. 27 c). it is noted
Chapter 7
Bar-type Traveling Wave Rotary Ultrasonic Motors
,,
12
W
10
~
8
c.
6
"" ,~ ~
211
- - - Phase A
_. -
PhaseB
?>
g
~
2
0 20
22
24
32
34
36
38
40
42
Freq llency/ kHz
Fig. 7.12
Velocity response versus frequency for stator
that the modal mass matrix M, is a positive number, hence the increase or decrease of the modified modal frequencies depends on the value of (K~ - wiM~) and modification ways. There are a number of ways to modify the modal frequencies. Compared with other adjustment ways. chamfering the stator is more convenient. In this section, how to define the position of chamfered portions is presented using finite element analyses. The finite element model of the stator is built using ANSYS software. The (K: - wiM:) value of the first bending mode is calculated utilizing the clement table, as shown in Fig. 7. 13.
Ant inode
- 243.068 - 167.20 1 -91334 - 15.467 - 205. 134 - 129.268 - 53.40 1 22.466
Fig. 7. 13
60.J
99.333
Distribution of ( K: - wkM:) value of first bending mode
Figure 7. 13 shows that the ( K:- wiM:) value of the area closed to the line passed through the antinode changes from positive to negative along the line from both ends to the middle. Therefore. it is effective to increase the modal frequency by removing the part of mass on both ends. It is also effective to decrease the modal frequency by removing the part of mass on the middle. However, the mid-
212
Ultrasonic Motors Technologies and Ap plicalions
die of the stator is piezoelectric ceramIc nngs. We can't modify this part. In fact. when the mass elosed to the middle is removed. it is also helpful in decreasing the modal frequency . If the regions ncar the line passed through the node are concerned as shown in Fig. 7. 13, the ( K~-w;M~) value doesn't change significantly from both ends to the middle. In particular, the ( K; - w;M~) value of the region between two slots of the stator is elose to zero. If the mass of these regions is removed, the modal frequency docs not change significantly. BTRUM operates using two orthogonal bending modes. If one point is located in the antinode of one of modes, the point must be located in the node of the other mode. Thus. when the regions near the line, which passes through the antinode of one of modes. arc modified, and the modal frequency of the mode can be changed. Meanwhile, the other modal frequency changes very little.
2. Example of modal frequency modification Modal frequency modification will be carried out for the stator mentioned above. The modal frequency induced by phase B is higher than that by phase A. Therefore. in order to adjust the modal frequencies to achieve the same. it is necessary to reduce the modal frequency corresponding to phase B and keep the modal frequency corresponding to phase A unchanged as much as possible. Chamfered portions on the stator are shown in Fig. 7. 11. The chamfered planes are perpendicular to the nodal diameter of the mode corresponding to phase A. In this way, the modal frequency to phase B can be decreased and that to phase A changes slightly. Fig. 7. 15 shows the velocity response versus the frequency of the modified stator. The solid line and the dash-dot line indicate the frequency response curves to phase A and phase B. respectively. The two modal frequencies of the stator arc 26. 23kHz and 26. 30kHz, respectively, which have the difference of 70 Hz that the original value is 170 Hz. After the stator is modified, this meets the requirements for the design. The efficiency, running stability, and output performance of the motor can be improved. II
10 ~
E
9
~
8
'" .E
g.
6
C
'"
4
·0 0
~
- - PhaseA
7 5
_ . - PhaseB
3 2 I
0
Fig. 7. 14 on stator
Portions modified
28 30 32 FrcquencylkHz
34
36
38
40
Fig. 7. 15 Velocity response versus frequency after modified stator
Chapter 7
7.3.7
Bar-type Traveling Wave Rotary Ultrasonic Motors
213
Design of Flexible Rotor
For BTRUM, the consumption of the input power can be divided into three parts L38 - : the first part is the damping loss in the conversion process from electrical energy to the mechanical energy. The second part is the friction loss between the stator and rotor, including their radial relative slip. This portion is 80 percent of the total energy 10ss L39- The third part is the inner loss of piezoelectric material. Thus, the key problem to improve the efficiency of BTRUM is how to avoid the relative slip between the stator and rotor. The two typical structure of rotors will be compared in the next section, and it can be proved that the rotor with the inner flange structure can effectively reduce the relative slip. The rotor with an outer flange structure is shown in Fig. 7. 16 (a). When the stator is in stationary state. the contact point on the stator and the point on rotor are assumed as points a and A. respectively. before the vibration of the stator is excited. In the xOz plane, when the stator is operating, the rotor is deformed by the action of the stator. In the condition, point a moves to a' and point A moves to A'. Because the directions of the displacements aa' and AA' are opposite in the x direction. the radial relative slip is inevitable.
I
zL' :
o
x
Enlarged local pari orthe map (a) Contact between outer HaJlge and the rotor
Rotor
zL "
O x .. ··•
Enl arged local pan oflhe map
(b) Contact between inner nange and the rotor
Fig. 7.16
Radial slip between stator and rotor
Reversely, for the rotor with an inner flange strueture. it ean be found from
214
Ultrasonic Motors Technologies and Ap plicalions
Fig. 7. 16(b) that the directions of the displacements of aa ' and AA' are identical in the x direction. and the radial relative slip is reduced. This conelusion can be further verified by finite element analysis. Two dimensional :'\Jode-to-Surface contact model is used to simulate the contact behavior between the stator and rotor. The element type is contac18. and the bending deformation is simulated by the synthesis motion of the stator in the x and z directions. Fig. 7.17 shows the results calculated by FEM. From Fig. 7.17(a). we can see that for the outer flange rotor. the direction of the displacements of the stator and rotor in the contact area are exactly opposite to each other in the :x; axis direction. and the relative displacement increases with time. From Fig. 7. 17 (b). we can sec that for the inner flange rotor, the displacements of the stator and rotor in the contact area possess the same direction. so the relative slip between the stator and rotor is reduced: 13J • This simulation gets the same conclusion as the above theoretical analysis does. X 10'"
X 10-7
3
2
~
g
il '" ~
•
0
o
'"""
Displacemen. of J)Qint A Displacemelll of point a
"
;6;
Displacement of point A
-I
o
Displacement of point a
<;
- 1.5
1l " ~
-2
'i5 -2.5
'i5
-.; '6
-.; '6 -2 c<
-0.5
"
c<
-3
-3 - 3.5
-4
0.2
0.4
0.6
-4
0.8
0.2
Fig. 7. 17
0.4
0.6
0.8
lI s
lI s (al Contact points between stator and rotor witll Ollte r fl ange
(b) Contact points between stator and rotor witll inner flange
Displacements of contact points on stator and rotor
7. 4
Performance Simulation for SDOF Motor
7.4.1
Dynamic Model
In order to enhance the wear resistance of a rotor, whose surface is coated with the friction material and its thickness is
o.
2-0. 3mm. Without taking into account
the deformation of the metallic base of the rotor in the contact process, the contact only makes the friction material layer deform. According to :'\J ewton's laws, the dynamic equations of the rotor can be expressed by the rigid motion in the axial direction z and the rotation motion in circumferential direction (3, that is
M,z+ C:z J,~+ C~~
=
Fi -
Po
(7.56)
Mri - Mr
(7.57)
=
Chapter 7
Bar-type Traveling Wave Rotary Ultrasonic Motors
215
where M, is the mass, ], the moment of inertia, C~ the damping in the z direction, C~ the damping in the circumferential direction, Fi the pre-pressure in the z dircction from thc stator, MTi thc driving torquc from thc stator, Po the pressure in the z direction from the spring, and MT the inverse torque from the load. Combining Eqs. (7.13), (7.56), and (7.57), the dynamic model of the motor can be written as
~.:j +Cq~j
i M,z+ q2
+w: qjq, :-
+ Cqq, + w =
K22 VB
+~~f(qj
,q,
'~j
'.il2)
+ Fq (qj ,q, ,qj ,q,)
Fi - Po
C~z =
],~+ C~~
KllV A
(7.58)
MTi - MT
where Kll and K22 arc the force coefficients of phases A and B, respectively, C q is the damping coefficient, VA and VB arc the voltages applied to the ceramic rings corresponding to phases A and B, respectively. w is the bending frequency of the stator, and F~ and F~ are the modal forces, which are composed of the contact force in the radial, circumferential, and z directions, respectively. They arc decided by the relative deformation of the friction layer, the contact area, and the relative velocity between the stator and rotor. Their expressions will be derived in the following contact analysis.
7. 4. 2
Contact Analysis
The driving part of the stator is divided into eight hexahedron elements. In order to improve the calculation precision of the contact area, some assistant points arc added between two nodes as shown in Fig. 7. 18. The displacements of these interpolation points can be obtained from Eq. (7. 22). The interface force between the stator and rotor can be simulated by a set of linear springs.
N/+I
'c.......- - - - - --
End
surface of the rOlOr
Equilibri um position
Interpolation point s
Co mac. area
Fig. 7.18
End urface of the stator
Nodes of the element
Estimation of contact area
Assuming that the displacement of the rotor at time t is z(t) and the displacement in the z direction of the assistant point n i is wet). Only when wet) > z(t), the point will contact the rotor and the deformation of the friction layer can be expressed as g
=
{Wet) - z(t), 0,
w>z w~z
(7.59)
The contact force between the stator and rotor can be decomposed into the contact pressure in in the z direction, and the friction force id in plane (r,B).
216
Ultrasonic Motors Technologies and Ap plicalions
According to the Coulomb law, it is obtained that {
In
kng
=
Id =
(7.60)
/1dIn
where k n is the equivalent stiffness coefficient of the spring, and /1d is the dynamic friction coefficient. So the force vector that the rotor applies to the stator surfacc can cxprcssed as
(7.61) wherc the sign functions y, and Yo are dccidcd by thc relative velocitics bctwcen thc stator and rotor, f, and fa arc thc componcnts of fd in thc radial and tangcntial directions, respectively. In FEM, cxternal loads are always imposcd on thc elcmental nodes, so thc contact interface force applicd to thc stator must be cquivalent to thc nodes in thc contact area. According to the virtual work principle, the virtual work by the contact force and by the nodal force should be the same, that is
t5;;Fe
=
If t5
JT
fint dS =
t5;;IfN~fintdS
S
(7.62)
S
where S is the contact area, Fe the equivalent nodal load, 15;" the displacement of the node, and N m the shape function matrix.
Fe
ffN;~JinldS
=
(7.63)
S
According to the virtual work principle, the corrcsponding modal forcc can bc also derivcd from
(7.61) Thcrefore, the modal force can bc cxpressed as Fq
= t»
TII Tf dS Nrn
iut
(7.65)
S
In the contact area, taking the integration of the contact interface force in the z direction, Fi in Eq. (7.58) can be
Fi
=
fflndS
(7.66)
s
Similarly, thc driving torquc can be expressed as MTi
=
IfYafardS
(7.67)
s
where r is the driving radius.
7.4.3
Performance Simulation
Based on Eq. (7.58), the mechanical performance of the BTRUM with 20mm in diametcr has been simulated.
Chapter 7
Bar-type Traveling Wave Rotary Ultrasonic Motors
217
When the pre-pressure is 60:'\1, Fig. 7. 19 shows the simulation results of the response characteristic. The motor can reach the stable speed 292r/min after 1. 75ms when the driving voltage is 200Vpp , the pre-pressure is 60:'\1 and the driving frequency is 31. 3kHz. Before the speed becomes stable, the maximum fluctuation of the speed is 12r/min. and then reduces to 3r/min. Fig. 7. 20 shows the start-up response characteristic of the motor under different loads. When the load is larger. the start-up time will be longer and the stable speed will reduce. 350 300 -§ {
250
'B "" ir
200
~ <5
100
150
50 1.5
2,;
2
I/ ms
Fig. 7.19
Start-up response characteristic of BTRUM without load 500
-=
i
Without load
-
400
.. .. .. 0,05N' III •. _.- 0,12N · III
300
ir
E-
<5
,
0
0
0.5
1.5
2
2,;
3
3,;
4
4 ,;
5
I/ ms
Fig. 7. 20
Start-up response characteristic under different loads
As shown in Fig. 7. 21. the load performance is simulated when the driving voltage is 300Vpp and the pre-pressure is 70:'\1. The maximum rotation speed is 240r/min, the stall torque is o. 146:'\1-m, and the output power reaches o. 92W when the output torque is o. 073N-m. The performance of the prototype motor is also measured and also expressed in Fig. 7. 21. The measured maximum speed. stall torque. and maximum output power are 230r/min, o. 153N-m. and o. 7W. respectively. Compared between the experimental results, it can be found that the model gives a satisfactory precision. The relationship between the speed and the torque under different pre-pressure is simulated. as shown in Fig. 7. 22. When the pre-pressure increases. the output speed of motor will be decreased, and the stall torque will be increased. At the same time, with the increase of the pre-pressure, the mechanical characteris-
Ultrasonic Motors Technologies and Ap plicalions
218
250
1.2
Expcrimcllll
0
200
." -E
]'
~
g-
6
+ 0
0.8
150
+".
+
0
100
0.6
+
o
0.4 0
50
0.D2 0.04
0.06 0.08
O. I
~ "0~
Q.
B0
0.2 0 .12
0. 14 0.16
Output torque/eN . Ill)
Fig. 7. 21
Load performance of BTRUM
tics will change from" soft" to "hard". which means the driving ability of the motor will be enhanced. 300r---~----'-----r---~----,
- - 70N
250
···· ···· 90N _._._ . l iO N
§ '00
i ~50 ~
100
~ 6 50 0
".:' .. 0
0.04
0.08
0.12
0. 16
0.2
Output torq ue/(N ' 111)
Fig. 7. 22
Speed vs. torque under different pre-pressures
The relationship between the output power and the output torque under different pre-pressures is also simulated, as shown in Fig. 7. 23. It can be seen that, as the pre-pressure increases, the output power will increase at first and then decrease. That means the output power will reach its maximum at optimized pre-pressure.
- - 70N
1.6 :>
~
" ~
·······90N _._._ . l iO N
1.2 ." ,
...-. _..........'":: ...
"
' .'.
0.8
§.
o 0.4
' .,
"
\.
0.04
0.08
0. 12
0. 16
,., 0.2
Out put torque/(N · Ill)
Fig. 7. 23
Output power vs. torque under different pre-pressures
Chapter 7
7.5 7.5.1
Bar-type Traveling Wave Rotary Ultrasonic Motors
219
Motion Mechanism of 3-DOF Motor Construction and Operating Modes
The structure of BTRUM with 3-DOF developed by PDLab is shown in Fig. 7. 21. It is composed of a cylindrical stator and spherical rotor 32J • The stator is a Langevin vibrator. whose accessories including a head. upper part. mounting support, electrode. lower part, three groups of piezoelectric ceramic components (A, B, and C) and a trail part all clamped by a bolt. Taking into accoun t the eff ecti ve elliptical concept proposed in Section 7. 2. 2. the rotor is of the spherical shape. There are two approaches to apply the pre-pressure between the stator and rotor. One is provided by weight of the rotor, and another is provided by the magnetic components placed inside the stator. Rotor
Mounting support
(a) Configuration of motor wilh 3-DOF
Fig. 7. 24
Boll for applying Ih eprc-p ressu re (b) SlllIetural decompo il ion
Structure of the MDOF BTRUM
The 3-DOF BTRUM operates with the bending and longitudinal modes. When the excitation voltages with certain frequency, phase and amplitude are applied to A, B, and C groups of piezoelectric ceramic rings, three operating modes of the stator will be excited: two bending modes (a) and (b), one longitudinal mode (c). The elliptical motion will be produced on the driving surface by combining these three modes, as shown in Fig. 7. 25.
7.5.2
Motion Mechanism
Fig. 7. 26 is the sketch of the driving surface. A series of driving points exist in the contact area. Point 0 is the center of the circle and the position of any point P can be expressed by radius r and angle e. When the two second bending modes are composed, a traveling wave will be produced on the driving surface with the same operating principle as the SDOF BTRUM. The elliptical motion of the driving point, as shown in Fig. 7. 6. will drive the rotor to rotate around z axis.
Ultrasonic Motors Technologies and Ap plicalions
220
z
x (a)
xg".".,
~~
(a)+(c)
(c)
Driving surface of stator
Fig. 7. 25 Driving principle of 3-DOF BTRUM
Fig. 7. 26
Driving points
Driving surface of stator
When the bending vibration to phase A and the longitudinal vibration to phase C are composed, the coordinate of point P in .xOz plane can be expressed as {
Wo eosw]t + reose
xI'
=
zp
=-
r (30 eose eosw1t + Zo sinw1t
(7.68)
where Zo represents the amplitude of the longitudinal vibration. Taking the coordinate translation: {
XI'
=
Zp =
x'
+ reose
z'
(7.69)
Inserting Eq. (7. 69)into Eq. (7.68) yields {
.x'
=
z'
=-
Woeosw1t r (30 eose eosW1 t
+ Zo sinw1t
(7.70)
Eliminating the variable t in Eq. (7.70) can lead to the trajectory equation of point P in x'Oz' coordinate plane: (7.71) Equation (7.71) is an ellipse equation, and the angle rJ exists between the long axis of the ellipse and the x' axis, which can be expressed as rJ
=
1 ( 2 r (30 Wo eose ) -2 arctan , 2 ,2e+ Z' _ W' r (30 cos 0 0
(7.72)
The motion of point P in the .x'Oz' plane is simulated and shown in Fig. 7. 27. It can be found that the trajectory of point P is an ellipse when the bending vibration to phase A and the longitudinal vibration to phase C are composed. This elliptical motion will drive the spherical rotor to rotate around y a.xis. Similarly, Fig. 7. 28 shows the simulation results of the motion of 16 driving points on the surface of the stator. It can be seen that a standing wave will be e.rcited when the longitudinal and the bending vibration are composed, which can make the points on the driving sur face induce to composed motion of the extention-contractin and the swaying vibrations of the stator, then the rotor will rotate around the y axis. Similarly, the rotor will rotate around the .x axis when the bending vibration of phase B
Chapter 7
Bar-type Traveling Wave Rotary Ultrasonic Motors
221
and longitudinal vibration to phase C are excited at the same time.
20 2W()
10
Z'
2Z()
0
-10
-20 -20
-10
0 x'
10
20
Trajectory of point P composing bending and longitudinal vibrations
Fig. 7. 27
.: .......... ': ··,··,·.. ··.. ·...... 'T·.. ·· .. ···[................................ .
~ :. ; ~ ~
-
..... .
1- T/4
--- 1- 1'12
-.- 1= 31Y4
z .. -
.,
.-' .:.-
!Y.......... :::,......-:::...... :::,.~.:::......::..... ;::. ..... .::..... ;:: ...... ..:::._;;:: .. .. ..::.... ,:::: ..... . . : : ,~...-:::...... :::,.. ~ ..'
o (a) In
o
xr- dime n iOIla! coordinate system
x (b) In x; plane
Fig. 7. 28 Motion simulation of 16 driving points composing bending and longitudinal vibrations
7. 6 7.6. 1
Optimal Design of Stator of 3-DOF Motor Construction and Objective Function
As discussed in Section 7. 5. 1, according to the various design requirements, the
222
Ultrasonic Motors Technologies and Ap plicalions
first longitudinal and the second bending modes are selected as the operating modes of the stator. The frequency value of the two modes should be as close as possible. This is the basic design requirement for the stator, which promises the good performance of the motor. The piezoelectric ceramic rings to excite the longitudinal vibration should be placed in the node plane of longitudinal mode shape. However, the piezoelectric ceramic rings to excite the bending vibration should be also placed in the crest or trough plane of the second bending mode shape. In addition, the node planes of the two modes should be as elose as possible, which can form an available mounting position for the motor. According to the above design criteria of the motor, the stator is a cylinder with variable diameter, as shown in Fig. 7. 29. The sensitivity analysis of the structure has showed that grooving I on the stator can effectively increase the amplitude, and effectively regulate the second bending frequency. The grooving II can effectively adjust the first longitudinal natural frequency. The groovings I, II, and III are very sensitive to the node plane and the wave trough position of the longitudinal and bending mode shapes. In general, the variable diameter design must be proper to meet the requirements of modes of the stator. z
y
Fig. 7. 29
z
Meshing of stator
Figure 7. 29 shows the structure, dimension and meshing of the stator. The element type is of a three-dimensional hexahedral element with eight nodes. Based on the results of the sensitivity analysis, structural parameters R" R" h2 , L1 , and L, are selected as the optimal variables. In order to meet the design requirements of the motor, the following issues have to be taken into account when establishing the objective function:" <2J : (1) The first longitudinal mode frequency III , the second bending mode frequency II., , and the target frequency It should be as elose as possible, that is
Chapter 7
Bar-type Traveling Wave Rotary Ultrasonic Motors
Fj
I
=
fll -
ft I
+ I fb'
-
223
(7.73)
ft I
(2) The piezoelectric ceramic rings for the longitudinal mode should be placed in its nodal planc. Thc position of thc nodc planc of thc first longitudinal mode and that of the middle node plane of the second bending mode should be as elose as possible, which can be described by minimizing the summation of the absolute value of the two mode shape values of the layer 7, that is
F,
=
I
¢it I + I
(7.74)
¢~~ I
where ¢it and ¢~~ arc the mode shape values of the first longitudinal mode in z direction and second bending mode in the x direction, respectively. (3) The piezoelectric ceramic rings for the bending modes should be placed in their wave peaks, which can be expressed by minimizing the difference of the bending mode values in the :r direction of layers 11 and 12.
(7.75) where ¢i,l" and ¢i,;x are the second bending mode values in the .1': direction of layer 11 and layer 12, respectively. Because F1 , F 2 , and F, are dimensionless quantities and possess various orders of magnitude, they can be synthesized into an objective function
(7.76) where a1 ,a, and a, arc the corresponding weighted coefficients. Therefore, the mathematical model for optimal design of the stator is the minimization problem, namely mm
7.6.2
F
=
a1 F1
+ a2 F2 + a, F,
(7.77)
Optimal Algorithm and Results
In MATLAB, the constrained variable metric method algorithm is used to resolve this optimal problem- 13 -11J • The weighted coefficients aj' a" and a3 arc equal to 2 X 10- 5 , O. 25, and o. 5, respectively. The target frequency is 29kHz. We take o. 01 as the terminative tolerance for computing F. The material properties of the stator arc listed in Table 7. 5. Table 7. 5 Material
Material parameters of the stator
Density p
Young's modulus E
/Ckg/m')
/C'-'/m')
Poisson's ratio f.1
7 550
82 X 10 9
0.22
Head part Cstecl15:f::)
7 860
209 X 10 9
0.30
Other parts Cbrass)
8 960
98 X 10
0.267
Piezoelectric ceramics
9
The stator of the new prototype of the 3-DOF BTRUM (20mmX 56. 9mm)is designed and fabricated by using the optimized parameters. The operating modes of the stator arc measured with PSV-300F-B. The comparison between the theo-
Ultrasonic Motors Technologies and Ap plicalions
224
retical and experimental results is shown in Table 7. 6, Figs. 7. 30 and 7. 31. Table 7.6
Modal frequency of the stator
Optimal result
Experimental result
Tolerance/ %
fll/Hz
28 127
27 410
2.5
.hz/Hz
28 711
27120
1.5
crror/ %
2.0
0.0
I
~~~~r-'----r'-~rTF---,-----~
0.8 0.6 0.4
8. 0.2 0
~
"
"'0
~
-0.2 - 0.4
-0.6 -0.8 -I oL----oJ.o~ILJ~ 0.L 02--~Ou.OL3---L OL 04L---O~.OL5----0J .~ ~
Length of the stalor/m
Fig. 7. 30
(al First order longItudinal mode shape
Fig. 7. 31
Calculated mode shape
(b) Sec.ond order bending mode shape
Experimental mode shape of the statorCalong axis direction)
Based on the analysis and comparison of the results, the following conclusions can be drawn: (1) The experimental results, including the frequency and the mode shape, are all similar to the theoretical results. (2) It can be seen from Table 7. 6 that the second bending mode frequency is close to the first longitudinal one. So the coincidence requirement for the operat-
Chapter 7
Bar-type Traveling Wave Rotary Ultrasonic Motors
225
ing frequencies has been met. (3) From Fig. 7. 31 (b). we can sec that the amplitude of the driving surface of the stator (on the left) is comparatively larger and the two groups of ceramic rings for the bending vibration are basically located in the crest plane of the bending mode shape. (4) From Figs. 7. 30 and 7. 31, it can be found that the nodal plane of the longitudinal mode coincides with the middle nodal plane of the bending mode. The mounting position of the motor and the ceramic rings' position for longitudinal mode are all nearby the same nodal plane of the stator.
7.7
Performance Measurement of 3-DOF Motor
7.7. 1
Testing Equipment and Results
In order to test the output performance of the prototype motor, an equipment is developed as shown in Fig. 7. 32. The stator is mounted on the skateboard, the rotor and its shaft arc fixed on the bearing. the skateboard 2 can move upward or downward through tuning the bolt to change the deformation of the spring and adjust the pre-pressure between the stator and rotor. In addition. the load can be applied to the shaft of the rotor through the additional structure composed of the chain wheel and poises. Integrating this equipment with the driver and the measurement system for USM load characteristics (see Chap. 11), the output speed and torque of the motor can be measured automatically.
Bearing support Rotor Axis
Cilain wheel z
bx
y 0 Stator
L»t~I---.-Ei----l-
Spring
- Poise Bolt
(a) Rotati ng aro und horizontal axis x (y)
Fig. 7. 32
(b) Rotati ng around ve rtical axis=
Testing device for mechanical performance of 3-DOF BTRUM
The output performance of the prototype motor is shown in Table 7. 7. under the pre-pressure of llN. excitation voltage of 300VI'I" and driving frequency of 27. 4kHz.
Ultrasonic Motors Technologies and Ap plicalions
226
Table 7.7
Output performance of the motor Maximum output value
Rotation
Speed without
Stall torque
aXiS
load/ (r / min)
/(mN'm)
Torque /(m"f'm)
x
33 35 66
21.78 36.57 30.20
12. 61 22. 80 13.78
y
z
7.7.2
Speed
Power
fer/min)
/mW
20 26
30 60 70
45
Effect of Pre-pressure on Mechanical Performance
When the motor is assembled, its output performance can be changed by the prepressure between the stator and rotor. So the relationships between the no-load speed and pre-pressure, and between the stall torque and pre-pressure are tested. The experimental results under the excitation voltage of 300Vpp and the driving frequency of 27. 4kHz. arc shown in Fig. 7. 33. 60
35
50
30
0
t:
~
0
40
E
Z 20 E
~
E" 30
"5'
"
0
~
0 Experimental data
20
"~
15
:;a
10
.9
Fitt ing clIrve
'"8. 10
0 Experi mental data
Fitting curve
ti;
5
(rJ
0
25
0
5
10
15
20
25
0
30
0
5
10
Pre-pressureIN
15
20
25
30
25
30
Pre- pressurelN (a) Rotating arOllnd ax is x 45
60 ~
.S
40
50
z~ ~
{ 40
~
"0
~ 0
30
-=
"
Fitting curvc
"0
30 25
~ 20 .9
0 Experimenta l da ta
~ 20
'"'" ~
0
35
ti;
10
0
15
Experi mcn tal data Filling curve
10 5
0
0
5
10
15
20
25
30
0
0
10
5
Pre· pres ureIN (b) Rotating arou nd ax is
Fig. 7. 33
15
20
Pre·pre u reIN
=
Relationship between the output performance and the pre-pressure
Figure 7. 33 shows that the pre-pressure affects the revolution speed and torque of the motor greatly. The suitable pre-pressure is more important to improve the output performance.
Chapter 7
7.8
Bar-type Traveling Wave Rotary Ultrasonic Motors
227
Driving and Control Techniques of 3-DOF Motor
An ultrasonic motor is actuated by the alternating voltages. Its motion can be adjusted by changing the phase, frequency, and amplitude of applied voltages. Therefore, the ultrasonic motor can be controlled by means of three basic methods: phase control, frequency one and voltage amplitude one. The frequency control can be easily realized, and can change the motor speed in the whole range, which is widely used for the control of traveling wave ultrasonic motor. But this method can not adjust the three driving frequencies of the 3-DOF BTRUM independently as well as the rotational speed in three directions. The voltage amplitude control can change the amplitude and thus change the output speed of the motor. Because the relationship between the voltage amplitude and speed is linear, the good control effect can be easily achieved even by using the traditional PID controller. In this chapter, this control method is used.
7. 8. 1
Configuration of the Control System
The motor control system is mainly composed of the motion controller, the driver, ultrasonic motors, sensors, computers and other components. The 4-axis motion controller GO-100 is produced by Shenzhen Gugao Company. Two optical encoders and a curved cross mechanism arc used as the sensors to detect the rotation angle around the x and y axes. The optical encoder YGM404 is produced by :'\Janjing Astronomical Instrument Development Center of Chinese Academy of Sciences. After the fourfold frequency processing, the angular resolution reaches 0.045°. The cross mechanism is shown in Fig. 7. 34. The 2-DOF motion control experiment can be realized in this system.
(a) Front view
Fig. 7. 34
7.8.2
(b) Vettical view
2-DOF BTRUM motion control system
Control for Trajectory Tracking
The block diagram of the control system is shown in Fig. 7. 35. The step-by-step interpolation and PID control strategy arc used to control the motion of the motor:" "J. PID controller is used to regulate the voltage amplitudes applied to the ceramic rings
228
Ultrasonic Motors Technologies and Ap plicalions
of groups A and B. and the voltage applied to the ceramic rings of group C maintains a constant amplitude. The step-by-step interpolation is applied to the position control.
Target trajcclOI'y
Fig. 7. 35
Block diagram of thc control systcm
U sing the prototype motor and the control systems. the motion control performance of the motor is tested. The experimental results arc shown in Fig. 7. 36. The error between the actual and target trajectories is less than 10 pulses, which means the control accuracy is 0.15°. 600 'n
;;
400
400
x
~
200
"""
0
0
200
'"
10: .................. ...... .. j....... :..r.::.:... :........ :... .....
""""i>-, . -"""'_
"0
e - 200
5
~
g 00 -400 .:
e
- 100 - 200
'~
-300
c<
-400 '----'----'--........---'----'~-'----'----' -400 - 300- 200 - 100 0 100 200 300 400
ell
- 600 - 600 -400 - 200
"0
0
200
400
600
.........
"13 co ~
'"
00
.c
· · ····~·--······l··
~
<> "13
'"
x ~
'x
.~
;;; <5
300
;;
0
.---r--~-'--"--'-'
0.,..
Rotating angel around axis .r/( XO.04So )
Rotating angel around axis>-!( XO.045°) (b) Circ ulartrajectory
(a) Linear trajectory 300r---,---,---,----.--~--_,
300 r-----------~-------~--~
;;
200
200
~
100
~
0",
0 X
············t
···········t
~
.~
"0
"e'"
a
'" -1 00 ], <:
'eJl" = '" <5
' ';:
'"
-200 -300 1..-.._ _' - -_ _' - - _"---_ -'--_ - ' - - - - - ' - 300 - 200 -100 o 100 200 300
- 300 I..-.._'--_'--_'--_'--_'------l - 300 -200 -1 00 o 100 200 300
Ro tating angel arou nd axis x/( X O.04So )
Rotating angel arou nd axis _~/( XO.04S·)
(c) Square trajectory
(d) Pentagonal trajectory
Fig. 7. 36
Experimental results of target tracking
Chapter 7
Bar-type Traveling Wave Rotary Ultrasonic Motors
229
References [ 1J
Hua Zhu, Chunsheng Zhao. The review of rotational ultrasonic micro-motor. Chinese & Acuustooptics, 2005, 27(6): 627-630,642. (in Chinese)
J our-
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[ 2J
T Morita, M K Kurosawa, T Higuchi. A cylindrical micro-ultrasonic motor using PZT thin film deposited by single process hydrothermal method (Diameter 2.1mm, LI0mm Stator Transducer). IEEE Trans. Ultrason. Ferroelectr. Frequency Control, 1998, 15 (5): 1178-1187.
[ 3J
T Morita, M K Kurosawa, T Higuchi. A cylindrical shaped mieroultrasonie motor utilizing PZT thin film (Diameter 1. 1mm and L5. Omm Stator Transducer). Sensors and Actuators A:
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Physical, 2000, 83 (3): 225-230. B Koc, S Cagatay, K Uehino. A piezoelectric motor using two orthogonal bending modes of a hollow cylinder. IEEE Trans. Ultrason. Ferroelectr. Frequency Control, 2002, 1 (19): 495-500. K Yoshida. Application of ultrasonic motor for conveying card mechanism. The 25"' Symposium in Research Institute of Electrical Communication. Japan: Tohoku University, 1989,
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S X Dong, S P Lim, K H Lee, et al. Piezoelectric ultrasonic mieromotor with 1. 5 mm diameter. IEEE Trans. Ultrason. Ferroelectr. Frequency Control, 2003,50(1): 361-367. Kai Zhang, Tieying Zhou, H uan Wang, et al. Study on piezoelectric cylinder micro ultrasonic motor with Imm diameter. Chinese Journal of Acta Acustica, 2004, 29(3): 258-261. (in Chinese)
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Tieying Zhou, Kai Zhang, Yu Chen, et al. A cylindrical rod ultrasonic motor with Imm diameter and its application in endoscopic OCT. Chinese Science Bulletin, 2005, 50 (7): 713716. (in Chinese)
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M Kurosawa, K Nakamura, T Okamoto. An ultrasonic motor using bending vibrations of a short cylinder. IEEE Trans. Ultrason. Ferroelectr. Frequency Control, 1989, 36 (5): 517-521.
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Shuxiang Dong, Shuxin Wang, Wenjiang Shen, et al. Rod-type ultrasonic mini-motor based on bending vibration mode. Chinese Journal of Acta Acustica, 1999, 24(2): 120-127. (in Chinese)
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Tieying Zhou, Kai Zhang, Cunyue Lu, et al. A piezoelectric micro ultrasonic motor using the
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metallic square bar as the stator. China Small Motor Technology Conference and E.rhibition. Shanghai: No. 21 Research Institute of CETC, 2005: 151-158. (in Chinese) Xiangdong Zhao, Changqing Liu, Chunsheng Zhao. Study on the motion mechanism of a piezoelectric ultrasonic motor using cylinder-bending vibration. Chinese Journal of Applied Mechanics, 2000, 17(3): 120-123. (in Chinese)
[l3J [14J
Yinhui Dong. Study on a Rocking Head Type Ultrasonic Motor. Dissertation for the Degree of Master. Nanjing: :'-Ianjing University of Aeronautics and Astronautics, 2003. (in Chinese) Chunsheng Zhao, Xianglin Ma. Rod-shape traveling wave ultrasonic motor. Chinese Invention Patent, CN03132161, 2003-07-03. (in Chinese)
[15J
Chunsheng Zhao, Hua Zhu. Rod-shape traveling wave micro ultrasonic motor. Chinese Invention Patent, ZL200410065703, 2004-11-12. (in Chinese)
[16J
Chunsheng Zhao, Hua Zhu. Rod-shape traveling wave ultrasonic motor. Chinese Invention Patent, ZL 200110065701, 2001-11-12. (in Chinese)
[l7J
Chunsheng Zhao, Hua Zhu. Cylindrical traveling wave ultrasonic motor with double rotor driven by single phase signal. Chinese Invention Patent, CN200110065700, 2001-11-12. (in Chinese)
[18J
T Amano, T Ishii, K Nakamura, et al. An ultrasonic actuator with multi-degree of freedom using bending and longitudinal vibrations of a single stator. 1998 IEEE Ultrasonics Symposi-
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[l9J
[20J
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[25J
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[27J
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Ultrasonic Motors Technologies and Ap plicalions urn. Miyagi, Japan: the Ultrasonics, Ferroeleetries &. Frequency Control Society, 1998: 667670. K Takemura, :'-I Kojima, T Maeno. Develoment of a bar-shaped ultrasonic motor for three degrees-freedom motion. Proc. 1'" International Conference on Motion and Vibration Control. Zurich: Robotics and Automation Society, 1998: 195-200. K Takemura, T Maeno. Characteristics of an ultrasonic motor capable of generating a mutidegrees of freedom motion. Proc. IEEE Int. Conj". Robotics and Automation. San Francisco, California: Robotics and Automation Society, 2000: 3660-3665. K Takemura, D Harada, T Maeno. A master-slave system using a multi-DOF ultrasonic motor and its controller designed consider measured and simulated driving characteristics. Proc. 2001 IEEE/RS] Int. Conf. Intelligent Robotics and Systems. Hawaii, USA: ICRA/IROS Organizing Committee, 2001: 1977-1982. K Takemura, T Maeno. Design and control of an ultrasonic motor capable of generating multi-DOF motion. IEEE/ ASME Transactions on Mechatronics, 2001, 6 (4): 499-506. lunbiao Liu. Study on the Single Stator Ultrasonic Motor with Three Degrees Freedom and Its Control Techniques. Dissertation for the Degree of Doctor of Philosophy. Nanjing: :'-Ianjing University of Aeronautics and Astronautics, 2001. (in Chinese) Fengjiang Zhan. Study on the Control Techniques of Column-sphere Ultrasonic Motor with Three Degrees Freedom. Dissertation for the Degree of Master. :'-Ianjing: :'-Ianjing University of Aeronautics and Astronautics, 2003. (in Chinese) Zhirong Li. A Fundamental Study on the Cylinder-sphere USM with Multi-degree of Freedom and Its Control Techniques. Dissertation for the Degree of Doctor of Philosophy. :'-I anjing: :'-Ianjing University of Aeronautics and Astronautics, 2001. (in Chinese) lunbiao Liu, Weiqing Huang, Chunsheng Zhao. Study on the motion mechanism of a cylinder-sphere ultrasonic motor with three-degree of freedom. Chinese] oumal of Mechanical Science and Technology for Aerospace Engineering, 2002, 21(1): 609-611. (in Chinese) Fengjiang Zhan, Zhirong Li, Weiqing Huang, et at. Research on motion trajectory control of 3-DOF ultrasonic motor with single stator. Chinese] ournal of Vibration, Measurement & Diagnosis, 2002, 22 (1): 265-269. (in Chinese) Zhirong Li, Weiqing Huang, Chunsheng Zhao. Recent advances in the research of ultrasonic motors with multi degree of freedom. Chinese] ournal of Vibration, Measurement & Diagnosis, 2003, 23(3): 161-161. (in Chinese) Zhirong Li, Weiqing Huang, Chunsheng Zhao. On design of a cylinder-sphere ultrasonic motor with multi-degree of freedom. Chinese] ournal of Mechanical Science and Technology, 2004,23(2): 157-160. (in Chinese) Chunsheng Zhao, Zhirong Li, lunbiao Liu, et at. Optimal design of the stator of the cylindersphere ultrasonic motor with multi-degree of freedom. Chinese] ournal oj Piezuelectric'i and
Acoustooptics, 2004,26(1): 13-16. (in Chinese)
[31J
[32J
[33J [34J
[35J
lunbiao Liu, Weiqing Huang, Chunsheng Zhao. Development and application of spherical ultrasonic motor with multi-degree of freedom. Chinese] ournal oj Vibration Measurernent & Diagnosis, 2001, 21(2): 85-89. (in Chinese) Zhirong Li, Weiqing Huang, Chunsheng Zhao. Recent advances in the research of ultrasonic motors with multi degree of freedom. Chinese] ournal of Vibration Measurement & Diagnosis, 2003, 23(3): 161-164. (in Chinese) Junbiao Liu, Chunsheng Zhao. 3-DOF ultrasonic motor with single stator. Chinese Invention Patent, ZL02113163. 5,2002-06-11. (in Chinese) Weiqing Huang, Xianglin Ma, Yinghui Dong, et at. Investigation of motion mechanism of a bar-type traveling wave ultrasonic motor. Chinese] ournal of Vibration and Shock, 2001, 23 (2): 48-51. (in Chinese) Zheng Tao, Yinhui Dong, Chunsheng Zhao. Research on optimum place of piezoelectric ceramic plate in rod-type ultrasonic motor. Chinese] ournal of Piezuelectric'i
& Acoustuuptics,
2004,26(1): 20-23,24. (in Chinese) [:l6J
Hua Zhu, Chao Chen, Chunsheng Zhao. Investigation on a cylindrical ultrasonic mieromotor.
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Chinese Proceedings of the CSEE, 2006, 26 (12): 128-133. (in Chinese)
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Yingjie Lei, Shanwen Zhang, Jiwu Li, et al. Genetic Algorithm Toolbox for Use with MATLAB. Xi'an: Xidian University Press, 2005. (in Chinese)
[38J
Yinghui Dong, Zhirong Li, Chunsheng Zhao. Optimal design of the rotor and stator of a miniature bar type ultrasonic motor. Chinese] uurnal of Vibratiun, Measurernent & Diagnusis,
[39J
2002,22(3): 175-178. (in Chinese) M Aoyagi, N Yuuiehi, Y Tomidawa. Rod-type ultrasonic motor using two degenerate second flexural vibration modes and characteristic consideration using its equivalent circuit expres-
[10J
[11J
[12J [43J [11J [45J [16J [47J
sion. lapaneselournal of Applied Physics, 1995, 31: 5292-5297. Heming Sun. Study on Hybrid Transducer-type Ultrasonic Motor Using Longitudinal and Torsional Modes. Dissertation for the Degree of Doctor of Philosophy. Nanjing: Nanjing University of Aeronautics and Astronautics, 2000. (in Chinese) Huaihai Chen, Chuanrong Zhou. A structural dynamic design method with multi-order given frequency and positions of mode shape node lines. Chinese lournal of Applied Mechanics, 1996, 13 (1): 59-63. (in Chinese) Chunsheng Zhao, Zhirong Li, Weiqing Huang. Optimal design of the stator of a three-DOF ultrasonic motor. Sensor and Actuator 11.: Physical, 2005, 121 (2): 494-499. Lingxi Qian. Optimization Design of Engineering Structure. Beijing: China Water Power Press,1983: 71-81. (in Chinese) S P Han. A globally convergent method for nonlinear programming. lournal of Optimization Theory and Applications, 1977, 22(3): 297-311. Chao He. Automatic Control for Mechanical and Electrical System. Beijing: China Renming University Press, 2001. (in Chinese) Qinyao Zhu, Zhongxin Wei, Runxiao Wang, et al. Digital Control for Machine Tool. Beijing: Aviation Industry Press, 1993. (in Chinese) Yonghua Tao, Yixin Yi, Lusheng Ge. NewPID Control and Its Application. Beijing: China Machine Press, 1998. (in Chinese)
Chapter 8
Ultrasonic Motor Using LongitudinalTorsional Hybrid Vibration The ultrasonic motor using a longitudinal-torsional (L-T) hybrid vibration (L TUM) is a significant type of ultrasonic motor. The longitudinal and torsional vibrations are combined to generate the elliptic motions of points on a driving surface. The rotor is driven by the friction force between the stator and rotor. The tangential and axial direction vibrations of the point on the driving surface of L TUM can be excited by longitudinal and torsional piezoelectric pieces, respectively. In this way, the load characteristics of the motor can be controlled independently by changing two input voltages. One controls the rotary speed of the rotor, and the other controls the friction which transmits the driving force. While the motor operates, the contact area between the stator and rotor covers the whole end surface of the stator, which allows the motor to yield greater output torque. In general review, LTUMs can be divided into two main types: the first is a multi-mode type, which has the longitudinal and torsional vibrators in the stator. The second one is a mode conversion type, in which only one longitudinal vibration mode is involved, and the torsional vibration can be induced by the longitudinal vibration. LTUM can be applied to robots, saloon cars, household appliances, pinpoint devices, spacecrafts, and especially, where large torque at low speed is needed. Based on the research on LTUM in PDLab, this section expounds the movement mechanism, structural design and dynamic analysis of several LTUMs.
8. 1
Current Research on L TUM
The research on LTUM[12: can be traced back to the 1980's. In 1991, Ueha developed a type of L TUM which took advantage of the large deformation of the stack vibrator in combination with the torsional vibrator- 3J , as shown in Fig. 8. 1. The multi-layer type vibrator can generate larger displacement under a low voltage and non-resonance state. The operating frequency coincidence of the two vibrators can be ignored in the motors. The stator of the motor is 20mm in diameter. The pre-pressure applied to the stator is 100N, the voltage applied to the torsional vibrator is 25 0 V,,,,, , the voltage applied to the stack vibrator is 7 V,,,,, , the rotation speed of the motor without load is 196r/min, the stall torque is O. 3)J· m, and the maximal efficiency is 21 %. In 1994, U eha used the sandwich piezoelectric stator to substitute for multi-
C. Zhao, Ultrasonic Motors © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011
Chapter 8
Ultrasonic Motor Using Longitudinal-Torsional···
233
layer piezoelectric stator and developed a type of LTUM with 80mm in diametcrL' J , as shown in Fig. 8. 2. The performance of the motor is enhanced due to the higher pre-pressure, and the stall torque of the motor is as high as 16)J om. In 1996, Ueha developed the other type of LTUM with 120mm in diameter: 5: . The stall torque of the motor reaches 10)J om, but the rotation speed is only 6 r / min.
r-....---- Lock nul
r-...:::="--1----Lock nut
1~~~=--- Spring
I::J:::::S:~---
Rotor
Spring ROlor
~~!I~~!I~-- Multi-layer PZT TorsinoaJ
Torsional vibration PZT
fS8,f-·fieBEBffi~-:8:::,:;3-'l- vibralion PZT
Fig. 8. 1 LTUM developed by Ueha in 1991
Fig. 8. 2 LTUM developed by Ueha in 1991
Since 1992, Tomikawa has carried out numerous studies on the main types of L TUMs:':, as shown in Fig. 8. 3. Vibrator with shrunken diameter (I)
"
~
C '0' S 1iJ "
~ ';;
II ~ II I
~ .D ~
"0
~
~
"'~"
~
m-------=III-Mode
Vibrator with shrunken diameter (2)
Multistep vibrator
(3)
One vibrator revolving type
(4)
Sandwiched rotor type
(5)
Coaxially combined type
(6)
L-Mode
Q(][]D[J Rotor
q(][[IJ Rotor
Fig. 8. 3
~!'i~ Torsional vibrator
Laminated vibrator
Main types of LTUMs developed by Tomikawa
234
Ultrasonic Motors Technologies and Ap plicalions
There are also some research achievements on LTUM in China. For instance, Jifeng Guo has developed the motor with SOmm in diameter L7J , as shown in Fig. S. 4. The maximal stall torque can reaeh 13)J ·m. Tieying Zhou has developed the motor with 32mm in diameter, as shown in Fig. S. 5, whose stall torque is 1. 7)J·m.
Fig. 8. 4 LTUM developed by Jifcng Guo
Fig. 8. 5 LTUM developed by Ticying Zhou
Sinee 2000, PDLab has studied L TUM continuously. Fig. S. 6 presents some typical L TUMs developed by PDLab: (a) the L TUM with 20mm in diameter[R: and its stall torque is o. 25)J·m; (b) the LTUM with 15mm in diameter: 91o : and its the stall torque is 3. 5N'm, and no-load rotation speed is 27r/min; (c) the L TUM with 20mm in diameter, in which birotors and single mode stator are used, and the stall torque can reach 1. 2SN'm, no-load rotation speed is 53r/min; (d) the mode-conversion LTUM with 15mm in diameter, of which the stall torque can reach o. 32N'm, and no-load rotation speed is 1 325r/min.
Fig. 8. 6
Four types of LTUMs developed by PDLab
Chapter 8
Ultrasonic Motor Using Longitudinal-Torsional···
235
At present, LTUM is still at the developmental stage. Industrialization and commercialization are still our goals.
8.2
Multi-mode Type LTUM
8. 2. 1 Motion Mechanism 1. Driving procedure Figure 8. 7 shows a sketch of multi-mode type L TUM developed by PDLab[ll: The motor takes advantage of L-T vibration to generate elliptic motion on the driving end, which can drive the rotor to rotate through a friction transmission. This section will expound the motion mechanism of the motor. The motor is composed of a rotor, stator with two sets of piezoelectric ceramic pieces for longitudinal and torsional vibration, and a base. When alternating voltages with certain frequency are applied to the two sets, respectively, the L-T hybrid vibration will be excited in the stator. The torsional vibration component will produce the rotary torque; the longitudinal vibration will control the friction force between the stator and rotor, synchronizing with the torsional vibration. The L-T hybrid vibration can realize the unidirectional revolution and mechanical output torque of the rotor.
p;i~~~~~1:f-PZT for Longitudinal
Ba e
Fig. 8. 7
Sketch of multi-mode type LTUM developed by PDLab
The driving procedure and motion state of the motor can be illustrated III Fig. 8. 8. Firstly it is regulated that when the stator vibrates in a longitudinal extension condition, whose displacement is positive, contrarily, it is negative; secondly, when the stator vibrates in a clockwise torsional condition, whose velocity is positive; contrarily, it is negative. One moving period can be divided into eight states. For CD ~ Ql) (first half-cycle), the stator is in the extension state, in which the stator contacts with the rotor, and the stator's torsional vclocity is in converse clockwise way (CCW). The stator transfers its torsional velocity in CCW to the rotor through friction between the stator and rotor, and then the rotor rotates in CCW. For Ql)~CD (second half-cycle) , the stator is in the contraction state, the stator separates from the rotor. By this time the torsional velocity is just contrary to the previous procedure, the torsional velocity of the stator in
236
Ultrasonic Motors Technologies and Ap plicalions
--'" Rotational direction of rotor {t Stator in contract ion state
~
:ll:
"
;;;
:;;
- -
-r
~
0
75
5
~
''';
x""
""
~
~
is ,;;;
Vi
;;;
Stator in extension tate Point P On surface of stator
•
Point on end surface of stator
o
Torsional vibration direction ofstator surface
-----
p
P-
¢
¢
f
¢
f
f
-"1-
CD
I
"
;;;
;;;
~
0
S
75
"t;
t:
"
""
S
is ;;; ,;;;
S u .:: ;;;
Vi
ffi
± I'
CD
CD
CD
- _¢oJ~L¢o_ 1f
-P
~
¢:J
~
~
CD
0)
(j)
(a) Dri vi ng p rocedu re
CD CD 0
(i) U.x
/l,r
CD (0 CD ® (i)
= 14 COS(u/ I I
0
lilf
li,T:=-
mVr si n(JJ 1 I I I I
0
II,
1"'~U, s:na>l I
0
I-
~I (b) Stretch-out view of response fu nction
Fig. 8. 8
Driving procedure in one cycle
:~
Chapter 8
Ultrasonic Motor Using Longitudinal-Torsional···
237
CW can not transfer to the rotor. But the rotor can continue to rotate due to inertia. In the second, third···, nth cycle, the above procedure will be repeated and the micro angular displacement of the rotor obtained in each cycle will be accumulated to form macro rotation of the rotor. Only when one phase of the voltage is antiphase, the rotor will rotate in CWo According to the driving procedure in one cycle of the motor shown in Fig. 8. 8(a) and the stretch-out view of the longitudinal displacement u< (t) = U< sinwt, the torsional displacement ux(t) =Uxcoswt and the torsional vcloeityitx(t) = - w Uxsinwt shown in Fig. 8. 8 (b), we can get the amplitude value of U x (t) , itx (t) and u< (t) in one cycle in table 8.1. In order to write simply, let U x 1. Table 8. 1
Uz
1 ~,fi 2
rr
,fi ~
rr
rr
3
2~4rr
3
4rr~rr
5
rr~1rr
5
6
4rr~4rr
6
7
2
0
o ~_,fi w 2
,fi -2w~-w
,fi
o ~_,fi
-w~-2w
_,fi ~-1
-,fiw~O
-1 ~_,fi 2
O~,fiw
2
2
-,fi~O 2
2
From
o ~,fi 2
becoming worse From contact state to crit-
0
ical state
2
,fi
_,fi ~-1
2w~w
2
,fi
1rr~1rr
o ~,fi
w-----Zw
-1 ~_,fi 2
~rr~2rr
,fi ~ 1
,fiw~O
-,fi~O
1
2
2
2
to
Contact state is
1 ~,fi 2
o ~_,fi
2
state state
Best contact state
2
2
critical
contact state
,fi ~ 1
,fi ~
Driving
Contact state
Uz
U.r.
O~~
1~2
Uz =
Contact state between stator and rotor and driving procedure in one cycle
wt 1
=
2
Stator separate [rom rotor gradually Stator separate from rotor completely Stator separate [rom rotor completely Stator still separate [rom rotor completely
CD~(2)
(2)~GJ)
GJ)~CD
@~G)
G:J~®
®~CV
CV~®
®~CD
2. Elliptic motion of points on the driving sur face In order to analyze the elliptic motion of points on the driving surface of the motor, the simplified modcl shown in Fig. 8. 9 is adopted. When the motor runs, the stator will generate the torsional vibration ux(t) around z axis, as shown m Fig. 8. 9(a), and the extension-contraction vibration uz(t) along z direction, as shown in Fig. 8. 9(b). Therefore, the vibration of point P can be expressed as Ux
=
Uxsin(wt)
Uz
=
U z sin(wt
+ rp)
(8. 1) (8. 2)
where rp is the phase difference between the longitudinal and torsional vibration, ' U z arc their amplitudes of point p, respeetivcly.
Ux
Ultrasonic Motors Technologies and Ap plicalions
238
-
P P'
: : : :: :
I'
:
:
(b)
Fig. 8. 9
1I, ~ U,s in (/l) I + 9')
Vibration response of multi-mode type LTUM
Eliminating wt from Eqs. (8. 1) and (8. 2), we can get 2
U,
u~
-
2 uxu ~
UxU, cosrp
+
2
.
u,
U; =
Sill
2
(8. 3)
rp
From Eq. (8. 3) , we can see that the motion locus of point P is an ellipse. Taking various numerical values for cp, we can get various loci, as shown in Fig. 8. 10.
(b)
(a)
(e)
(d)
(g) -TC
<-TCI2
0>q:»-1I/2
Fig. 8. 10
(e)
Motion loci of point P under various phase differences
When rp= 2 and rp= IT
IT 2'
the motion loci of point P can be simplified as an
ideal elliptic equation
1
(8. 4)
Chapter 8
Ultrasonic Motor Using Longitudinal-Torsional···
239
We note that the directions of the motion are opposite under the two phase diffcrences
({J= ;
and
({J= -
;.
Accordingly, wc can changc
({J
to control the di-
rection of the motion. In addition, we can control the output characteristics by changing thc phasc difference of two cxciting voltages, bccausc its long and short axes of the ellipse are related to the longitudinal and torsional vibration, respectively. When ({J= 0 or ({J= IT, the equation of motion track is u<
=± ~>x
(8. 5)
Whcn the motion locus of point P is a s traigh t linc, then thc motor will not rotatc.
8. 2. 2
Structure Design of Multi-mode Type LTUM
1. Design requirements In order to enhance the output performance and electro-mechanical transformation efficiency, the structure of the motor must meet following requirements: (1) Designing reasonably piezoelectric ceramic pieces to excite vibration modes needed. (2) Using the proper longitudinal and torsional modes. (3) Longitudinal and torsional modc frcquencics must bc the same to get higher hybrid amplitude. (4) Placing piezoelectric ceramic picces in proper locations to improvc thc cxciting efficiency. 2. Design of piezoelectric ceramic pieces[1217] In thc ultrasonic motor with multi-mode, the longitudinal ceramic pieccs arc uscd for cxciting the extcnsion-contraction vibration of the stator along thickncss direction (TE type), using the d 33 effect of the piezoelectric ceramic pieces. The torsional ceramic pieces are used for exciting the shear vibration (TS) of the stator, using d 15 effect of the ceramic pieces, which are polarized along circumfercncc by two methods: (1) Dividing the whole annulus PZT into several arc segments, polarizing evcry scgmcnt, taking off thc silvcr layer from them, and thcn combining them into one whole torsional PZT piece, the silver layer coating on the up and bottom surfaccs, as shown in Fig. 8. 11 (a). (2) First polarizing rcctangular PZT piecc, thcn cutting it into an arc segment, and finally combining several arc segments into onc wholc torsional PZT piece, as shown in Fig. 8. 11 (b). The method (1) is relatively simple. However, the maldistribution of electric field intensity will lcad to the polarization disproportionation of the PZT piecc. Thereby, it will influence the whole performance of the stator. Theoretically, the more arc scgmcnts, thc bctter performancc thc stator has. Considcring thc complexity of polarization techniquc, thc arc segmcnts can not be too more. Thc
240
Ultrasonic Motors Technologies and Ap plicalions
method (2) can ensure uniform polarization of PZT pIece, but the method is relatively difficult to manufacture. We can select one of the two methods from practical condition.
(a) First shaping by cutting, then polarizing
(b) F irst polarizing, then shaping by CUll ing
Fig. 8. 11
Methods of polarizing PZT pieces used for multi-mode type LTUM
The specific layout of thc two scts of PZT pieces is shown in Fig. 8. 12. Each set is composed of two or four pieces. and they are placed with "parallel con-
nection" to enlarge the amplitude. PZT pieces for longitudinal vibration
PZT pieces for torsional vibration
Ground
Fig. 8. 12
Assembling of PZT pieces for multi-mode type LTUM
As shown in Fig. 8. 13, taking one annular PZT piece as example, whose thickness is hI"
the inside diameter is 2a, and the outside diameter is 2b.
The PZT piece has been polarized along circumference, and we apply electric field along axial direction, then the torque generated on the two surfaces of the PZT piece is (8. 6)
The equation above indicatcs that thc output torque of thc PZT piece is in-
Chapter 8
Ultrasonic Motor Using Longitudinal-Torsional···
241
versely proportional to the thickness. In order to get large torque. the thickness of the PZT picce should bc as thin as possible. But if the piecc is too thin, it will bc fracturcd easily in thc assembling proccss.
V
t~
b
·7 :t : ....-------!~~ ::-:- :,·,,·, '., ': ,:{: ::: :, ~ . ':::,':, ': :' ., . ".,. ':. :::::,' .. _... ~ .- .. -
B::&eee+1
I
' 11'
t
Polarization direction
Fig. 8. 13
Geometric parameters of PZT piece for torsional vibration component
3. Design method of consistency of longitudinal and torsional mode frequenczes Sincc low ordcr modc of a stator is casy to excite, and high order mode wastes higher energy than that of low order mode and is subject easily to mode disturbance, the design of the stator has a requirement that the stator vibration velocity should be as largc as possible, and thc velocity is decidcd by both thc displaccment and the frequency. We have to select a mode to obtain the maximum product of thc displacemcnt and the frequcncy. In addition, wc must consider the consistency of the longitudinal and torsional mode frequencies of the stator. Generally, for a uniform bar type LTUM, its nature frequencies possess such an order: the first order torsional frcquency, the first ordcr longitudinal one and the second order torsional one. To ensure stator's two operating frequencies to bc closcd, if the first order torsional mode and thc first ordcr longitudinal modc arc adopted. then, thc stator must be designed to incrcase thc first ordcr torsional frequency or/ and reduce the first order longitudinal frequency. If the first ordcr longitudinal mode and thc sccond ordcr torsional modc are adoptcd. Thcn, the stator must be designed to increase the first order longitudinal frequency or/ and rcduce thc second ordcr torsional frcquency:]R]. Fig. 8. 11 shows the stator with shrunken diameter. Because shrunken diame-
ter exists, the first order torsional frequency has small variation but the longitudinal frequency is decreased, which will make the first order torsional and first order longitudinal mode frequencies be close. Fig. 8. 15 is the sketch of the stator with multistep. Fig. 8. 16 shows some results of the stator with multistep calculated by the finite element method. It can be observed that if r, is fi.red
and r, is changed, the first order longitudinal frequency will arise with the augmentation of r3 • and the second order torsional frequency will decrease with the augmentation of r,. Thereby the first order longitudinal and the second order torsional frequencies can be close.
Ultrasonic Motors Technologies and Ap plicalions
242
Fig. 8. 14
Relationship between shrunken diameter and mode frequency of stator L, mode
-t-/;1-=I :2r, 2r,
2r
bl2
a
Fig. 8. 15
r:;
-EBJ-
fL,
~~ ~""""",\ bl2
Fig. 8. 16
Sketch of
mode
T, mode
-v~~T;' mode .tT,>.tT,
-~-
Relationship between multistep
and mode frequency of stator
stator with multistep
Based on the analysis above, PDLab has developed one multi-mode L TUM with diameter of
o. 045m.
The difference between the first order longitudinal and
the second torsional frequeneies is
o. 3kHz.
This indieates that the stator with
multistep ean effectively make the first order longitudinal and the second order torsional frequencies elose.
4. Optimum design for amplitude of stator The amplitude of the stator is generally few micro, how to design the stator is of great significance to augment the amplitude of the end surface of the stator, then improve the mechanical performance of motors. According to the magnification theory of ultrasonic solid horn- 19J
,
the relationship between the stator's structure
and the amplitude of driving end is studied by Heming Sun and Chunsheng Zhao in detail under the first order longitudinal mode. Fig. 8. 17 (a) shows the finite element model of a stator. The material used for the part with larger section is the steel which has a higher acoustic impedance, whereas the material used for the part with smaller section is aluminum which has a lower acoustic impedance. Fig. 8. 17 (b) shows the first order longitudinal mode shape of the stator. From Fig. 8. 17 (b) , the displacement amplitude value at right end is much greater than that at zero point of coordinate
.T.
Thereby, the stator with multistep has really
the function to magnify amplitude. As a matter of fact, besides the stator with multistep, the shape of the stator can be made as exponential function type, conic type, catenary curve type, and so on.
In a word, the stator with variable cross-section and the material with different acoustic impedance can notably magnify the amplitude at the free end of stator.
Chapter 8
Steel
PZT
Ultrasonic Motor Using Longitudinal-Torsional···
Duralumin
~
243
I
I
...Y -30 (a) Finite element model
Fig. 8. 17
8.3
-15
o
15
30
x/mm
45
(b) Amplitude distribution along axial direction
Change of displacement amplitude with variation of cross-section
Contact Model between Stator and Rotor
Like the traveling wave ultrasonic motors, the contact model between a stator and rotor of L TUM is vcry important. From thc model, we can conduct the simulation to prcdict thc load charactcristics of L TUM. W c can also analysc that thc load charactcristics are influenced by the amplitudes of longitudinal and torsional vibration, the pre-pressure between the stator and rotor, the friction material, and geometrical and physical parameters. It is to be regretted that so far as it goes, we cannot find a satisfactory contact model, since Ueha and Tomikawa firstly proposed the contact model in 1993: 20J • Based on the model, Jifeng Guo, Shujuan Gong and Xiao Liu, et al. have developed a model- 21 - 22J which takes advantage of energy conservation principle in one period, and derives the expressions of the output torquc, contact duration and othcr parametcrs in 2003. Finally, they applied these expressions for simulation.
8. 3. 1
Modeling of Contact Interface
In this section, we citcd and improved the model developed by Refs. [21J and [22]. Whcn thc motor opcrates in a stcady state, an elliptical motion trajcctory of a point on the stator surface is shown in Fig. 8. 18. The point moves along A, B, C, and D. U means torsional displaccment of thc point, and U y mcans the longitudinal displaccmcnt of the point. Thcy can bc cxpressed as E
{
U
E : - UEsinwt Uzcoswt -
(8.7)
Uz
wherc U x and U z arc the torsional and longitudinal amplitudcs of thc point respcctively. In the steady state, the stator's vibration is transformed into the rotor by intermittcnt contacts bctwcen thc stator and rotor with a friction layer. The deformation of friction layer can be described by Fig. 8. 18, where point a indicates thc initial contact point between thc stator and rotor, and point b indicates the final contact point. The axial displacement of the initial contact point is U m , ({Ja and ({Jb indicate thc initial and final contact anglcs, and thc corresponding timcs are ta and t b , rcspcctively. Then the contact duration ({J is cqual to (({Jb - ({Ja)' in which the rotor is always in contact with the stator. The contact time during every vi-
244
Ultrasonic Motors Technologies and Ap plicalions
B
A
wt
f)
Fig. 8. 18 Elliptic locus(lcIt) and contact conditionCright) of a point on stator surface
bration cyelc is Tc= cp/w. At this momcnt, thc relationship bctwccnCPa and contact duration cp is cpa = (rc - cp) /2. Hence (8. 8)
Rcgarding thc rotor as rigid body, thc forcc acting on thc rotor along axial direction is shown in Fig. 8. 19, where Po is the pre-pressure applied to the rotor by the spring, and P is the impact force acting on the rotor from the stator through thc friction laycr, thc rotor mass is m. and thc vibration pcriod of thc stator is T = 2rc/ w. Thc impact forcc P in onc vibration pcriod can bc expressed as t
E (0, t a )
t E (t a , t,, )
(8. 9)
t E (t b • 1')
where kr = ErS/ hr is the equivalent stiffness of the friction layer; E[O S, and hr are the Young's modulus. contact area. and thickness of the friction material, rcspcctively. C
~
ROlation axi s
Pre· pres lire /'0
Rotor
~~~~22222S:~- Friction layer 1111 pact force P
Fig. 8. 19
Axial force acting on rotor
In fact. the simplification model shown in Fig. 8. 19 is similar to a single de-
Chapter 8
Ultrasonic Motor Using Longitudinal-Torsional···
245
gree freedom system which is made up of a rotor and a spring. The natural frequency of thc system is lower due to lower spring stiffncss k. and thc frequcncy of thc impact forcc acting on thc rotor from the stator is abovc 20kHz when thc motor operates in a steady state. therefore the rotor can be considered as immovablc obj ect approximately in axial dircction. That is. ignoring thc momcntum variation in the axial direction. the axial impulse to the rotor in one vibration period is
[cp - Po)dt
[mdv
=
=
(8. 10)
0
from which
f T Pdt
=
Po 2IT
(8. 11)
w
o
In one vibration pcriod. thc impact forcc is influcnccd by thc contact time ta to tb' namely by contact anglc rpa to rpb. Therefore Eq. (8. 11) can bc simplified as
f::Pd(wt)
=
2ITPo
(8.12)
Substituting Eqs. (8. 8) and (8. 9) into Eq. (8. 12). it can bc obtaincd
2ITP o
=
klU~
[2sin(rp/2) - rp cos(rp/2) ]
(8. 13)
Equation (8.13) indicatcs that thc contact duration rp is related to the prcpressurc. elasticity modulus. contact arca. thickncss of friction layer. and longitudinal amplitude value. If these parameters are given. the contact duration can be obtained. When rp=2IT. the stator contacts with the rotor during the whole vibration period. and the pre-pressure is kfU, all the time. When the pre-pressure is greater than kIU,. the stator will contact with the rotor always. klU, is a critical value between continuous and intermittent contact. so the critical pre-pressure is defincd as P,=kIU,. Therefore in Eq. (8. 9) the impact force can be revised as
Po < P, Po ;?: P,
= =
klUx klUx
(8. 14)
From Eq. (8. 7). thc tangcntial vibration velocity of point on stator surface is and V" is the velocity amplitude of torsional vibration. then we have
Ux '
(8. 15) At the same time. considering that the rotor has certain inertia moment and thc contact time is short. thc rotation spccd of thc rotor can bc supposcd to be a steady value V,. Fig. 8. 20 shows the relationship between the rotation speed and the torsional vibration velocity of the stator surface during one vibration period. It is observcd from Fig. 8. 20(a) that when the stator begin contact with thc rotor and thc torsional vibration velocity is highcr than thc rotary spccd in thc moment. thc friction forcc F m will do positive work. as curve abo It can be also observcd from Fig. 8. 20(b) that when thc torsional vibration velocity is lower than
246
Ultrasonic Motors Technologies and Ap plicalions
the rotary speed, the friction force Fm will do negative work, as curve ac and db. A friction coefficient is relevant to relative velocity between the torsional vibration velocity and the rotary speed, which is shown in Fig. 8. 21 and can be expressed as[23:
(8. 16) where /1d is a sliding friction coefficient. /1, is a static friction coefficient and k[ is the coefficient of adhesion, as shown in Fig. 8. 21.
(~ I
(~I
(a)
Fig. 8. 20
(b)
Relationship between the torsional velocity and the rotary speed of LTUM
Fig. 8. 21
Coulomb model
The total friction force between the stator and the rotor can be expressed as
Fm
=
ff,dS s
=
f/1
~ dS =
/1P
(8. 17)
s
Ignoring the friction moment induced by bearing and the damping moment caused by structure on the rotor. the torque produced in one vibration period should be the same as the torque consumed by the load, and the average output torque in one vibration period is
(8. 18) where Rov is the mean radius of the friction layer. In the case of Po < kIU~, there arc two situations: (1) when
Chapter 8
MT
=
~ krRvUz [~
Ultrasonic Motor Using Longitudinal-Torsional···
kl V,,(cp- sincp)
+
(k l V, - fl.d) (feos f
247
- sin f ) ] (8. 19)
(2) when CPI > cpa , the average output torque can be expressed as
If
I~' F mR,vdwt + -1 . FmR,vdw t
=
-1
=
MTB +Mm
IT
(8. 20)
r.
IT
r
Equation (8.20) divides the moment into two parts: one is a driving moment MID which is positive and the other is a brake moment MTB which is negative,
which is corresponding to forward and backward of point c in Fig. 8. 20 (b). At point c, CPI = arcsin(VjV,J Considering Po
~ kr
RvUz [
+ (k V, l
+ (k
l
V,
~
kl V" ( IT - 2cpj
fl.d) ( -
COSCPI
+ ;
+ sin2cpj cos f
.!£.2 + fl.d) (coscpj - Sill
4coscpj cos f
)
- CPI cos f ) ]
IT-P!£. 2 cos 2
+ cPj cos !£.)] 2
(8.21)
(8. 22)
Substituting Eqs. (8. 21) and (8. 22) into Eq. (8. 20), the output torque can be obtained:
(8. 23)
In the case of overpressure, Po?: Pc
CPI > cpa , thcn
MT
=
kl Rv [
~ V"P c -
V,Po
=
krUy ' there is only one situation of
]+ ~ Rv
fl.d [2P,COSCPI - 2PaCPI ]
(8. 21)
Equations (8. 23) and (8. 24) indicatc that there arc many factors which influence the output torque of the motor, including elastic stiffness of friction layer, longitudinal amplitudc, torsional amplitude and thc contact anglc. Assuming MT = 0 and substituting CPI = arcsin( Va /V,J into Eqs. (8. 23) and (8. 24), thc no-load specd Vo can bc dctermincd. When V, = 0 and cPj = 0, thc stall torquc can be obtaincd:
248
Ultrasonic Motors Technologies and Ap plicalions
1 k f R "V U y [ 11 k 1 V "rp ( - smrp " ) - /1d (!£ 2 cos,!£ 2 _.' sm !£) 2 ]
-;
(rpj
<
rpa ,P a
~
PJ
41rr k j PeR,v V" (rp - sinrp) +
M Tmax =
(rpl
>
rpa ,P 0
1 -ZkjPeR,Y"
~ /1dPeRv [2 -
sin {- - cos {- (rr - {- ) ]
~ PJ
2
+ -;/1dPeRav
(Po> Pc) (8. 25) When Po = pc. at critical contact duration rp= 2rr, and substituting it into the above equation, its stall torque is the same as that formed under condition of Po>
p,.. In which the stall torque is not related to the pre-pressure value. At this condi tion, the stall torque cannot be increased by enhancing pre-pressure.
8. 3. 2
Friction Loss on Interface and Efficiency of L TUM
Friction loss on the contact surfaces can be calculated as
(8. 26) where V; is the torsional velocity of the stator, V, is the linear velocity of contact point on the rotor. The efficiency between the stator and rotor can be expressed as P01lt
(8. 27)
where P oot is the output power
P OUI where
iJ is
=
1
T
IT MT pdt . 0
(8. 28)
the angular speed.
In the case of Po
<
kfUy' there are two situations:
(1) when rpl ~ rpa , from Eq. (8.26)
1 U < {k jV'" [. Pd = -;kf sm !£ 2 -
. 31 sm
3
!£ 2 - cos !£ 2 (!£ 1 + -.l.. 1 smrp ) ]
-k 1 VV V 2 (sin!£-!£cos!£) , "(!£--.lsinm)+k 2 2 T 1, 2 2 2 +/1dV,,({--! sinrp)+/1dV,({-cos{--sin{-)} (2) when rpl
>
rpa' the loss can be divided as driving and block parts:
(8. 29)
Chapter 8
=
Ultrasonic Motor Using Longitudinal-Torsional···
!L)_ (cosm
3 m -sin3 -.lkU 1( I y {k 1 V'"[-.l(eos 3 Tl 2
(2kl vy" + kl
V;, cos f
+,ud
-sin!L)] 2
!
V,,) [ (T - 1( ~ p)-
+ (kl V; + ,ud V, + 2kl V,V"cos f -
Tl
249
+ ,udV" cos f ) (sin f
(sin2rpl - sinrp) ] - COSrpl )
(,udV,cosf+klV;cosf)(rpl-1(~P)} (8. 30)
PdD
~ kfU
=
y
~
{kl V;, ( COSrpl -
cos 3 rpl )
(2kl vy" + kl V;, cos f
- ,udV,,) (
+ (kl V; -,ud V, + 2kl V,V"cos f + (,udV, cos f
~
-
T+ ! sin2rpl)
- ,ud V"COS f ) COSrpl
- kl V; cos f ) ( ; - rpl ) } (8. 31)
The total loss on the contact surface can be expressed as (8. 32)
8.3.3
Simulation of Performance of LTUM
From Eq. (8.25). we can predict the load characteristics and analyze the influence of the pre-pressure. exciting voltage and friction layer on the performance of the motor. Various parameters adopted in the simulation are shown in Table 8.2. Table 8.2
Parameters adopted in the simulation
Parameters
)J umerical values
Rotor Quality m/kg
2.0c-2
Pre-pressure Po / N
1. 2c2
Operating frequency
f /KHz
5. 2el
Elasticity modulus Er/Pa
3.5e-1O
Friction layer area 5/m 2
8.2c-5
Effective contact radius R,"/m
6.6e-3
Thickncss of friction layer hr/m
o.
Longitudinal vibration amplitudc Uy/m
3.0c-6
Torsional vibration amplitude Ux/m
3.0e-6
3e-3
1. Contact pressure between the stator and rotor The most essential influence for the output torque is the contact pressure between the stator and rotor. which is relevant to the longitudinal amplitude. pre-
Ultrasonic Motors Technologies and Ap plicalions
250
pressure and elasticity modulus and thickness of the friction layer, as shown m Figs. 8. 22-8. 25.
~::l ~
0.
~
8
550
400 ,-----------------------, :...... \ - - U,= 1e-6m /'"", .... _.__.... U,=2e-6m '1. \' 320 - - - U.=3e-6m .... ........ U,=4e-6m 240
/ ] \ ,. I /. '. \.
440
1
,,,~;./ ,:, ,,,,\~i ' q:
330
.~
;
,~
/. 160 , l,
~
' i\
13
220
.s
"0 U
,I .!
80 1/
- - P()=40N ........... P()=110N - - - P()=180N P()=250N
110
. ..... .
OL---~--~~~--~--~
o
0,2
0.6
0.4
0 ,8
0.2
0.4
Iff
Fig. 8. 22 Contact pressures vs. contact duration under various U z
i
~ 0.
;;
;:l o u
640
400
480
. ....
; , / ;.... ' \ \
;
:
.
320 ..... J .
...
:
-"r-0.3e-311l •.•..•..•.• " r=0,6e-311l
320
E,- 1.75e9
_ . _ E,=2.45e9 .
;/.- \ ,\. :,. .\ ':
,
- - E =0.35e9 --_._... E,= 1.05e9
....... 1'"\~ ''' : '' --;. -..'.
~
..
~ c-
..
:
- - - il r=0,ge-3m
240
o
;:l 160 0 u 80
0.2
0.8
Fig. 8. 23 Contact pressures vs. contact duration under various Po
800 .-------~--------~----,
./ .
0.6 tiT
0.6
0.4
0 0.8
""',.,',
0
..... .
0.2
111'
0.4
0.6
"., if
0,8
tiT
Fig. 8. 24 Contact pressures vs. contact duration under various Ef
Figures 8. 22-21 indicate that as U z
....
Fig. 8. 2S Con tact pressures vs. contact duration under various h f '
Po , and E f increase, the contact duration
will decrease and the amplitude of contact pressure will increase.
Fig. 8. 25
shows that as h f increases, contact area will increase and the amplitude of contact pressure will decrease.
2. Influence of pre-pressure on output characteristics Relationships of the stall torque and no-load speed versus the pre-pressure are shown in Figs. 8. 26 and 8. 27. The figure indicates that the stall torque will increase by the augmentation of pre-pressure, while the no-load speed will decrease. When the pre-pressure is higher than that at critical pre-pressure, the stall torque will no longer increase.
Chapter 8
0.6
Ultrasonic Motor Using Longitudinal-Torsional···
1500
.---~-~--~-~------,
0.48
251
r--~--~-~--~----'
1300
I
~.
~ "go .9
~
0.36
gj f}
0.24 . ..... .
1100 900
en
700
160
240
400
320
500~-~--~-~--~-~
o
120
Pre-pressureIN
Fig. 8. 26
240
360
600
480
Pre-pressure/N
Stall torque vs. pre-pressure
Fig. 8. 27
No-load speed vs. pre-pressure
Figure 8. 28 shows the load characteristics under various pre-pressures. It can be observed that when prc-pressure increases, the stall torque increases obviously, 1400
20r-----------------,
.. 1120 ~-...;..
. . <..;
~
......
840
:;;~
0-
'"
'"..
.
,
.
-
.
· ....••••• ~S.·0 ~ >. \ .\~ 0 IIlCre
16
- . _.
. .....·<~~'
~
::
- - l'o=40N : .......... I'D- li ON . .... : .... - - - " o- 180N l'o=250N
~\
-~
12
: \
...
o
0. 12
0.24
\
\ ~\~.~ . ~.....
036
0.48
0.6
. ...
-........... - - - . _ .
64
l'o=120N l'o= 160N l'o- 200N "o-240N
\
I .~,/
/1 /
'
32
-
: ~/
~ ,/~.(., ~~--;;~
..,r' .
~~
:
.. . . . . . . . .
:.
o
0. 11
:
~
:.
Fig. 8. 28
0.22 0.33 0.44 Output torquc l(N' m)
0.55
•
_ . - l'o- 240N
40
:.
0 2. 4 0.36 0,48 Output torque I(N' 111 )
'.
100 .------------------------,
•
20
O~--~--~--~~--~--~
0. 12
.
60 .. ..... : .... ... . ;... . . ...:.
16
o
\
O L---~--~--~~--~'--~
/:
48 ........:........ ; .. . ......:.:," . ,/. .; .. Poin,crease ..:. _., -"
\
- - " 0- 120N . ........... l'o= 160N 80 ... ,," .;"."". : "" - - - l'o=200N
I
.
\ \
\ : \ \ '\ .. \ .\ \.
OutputtorqllC I(N' m) 80 .-------------~---,
\ . \ \
.....\ .. ~ \ .\ .... \ : \ . .
~ .\ ',
0 L - _ - l l_ _~_~ , ~ · _ _L-~~
\ :\
\,
" : Po increase
.... :\ .:.. \
"
.
... .t, •••\ \ •••••••
\
............... \~. , .. ;........ :...... .
560
\ .
0.6
O L--~-~---"-----'~"--''---'
o
0. 12
0 2. 4 0.36 0.48 OmpUl torque I(N . 111)
Load characteristics under various pre-pressures
0.6
Ultrasonic Motors Technologies and Ap plicalions
252
but the no-load speed will decreases. and the load characteristics change from hard to soft. Whcn pre-prcssure is smallcr, thc torque is small and the spccd dccreascs sharply. Conversely. whcn pre-pressurc is largcr. thc torquc is large and thc speed decreases smoothly. On thc othcr hand. thc prc-prcssure has an cffcct on the output power, friction loss and efficiency: when pre-pressure increases, the output power will remain basically unchanged, but the friction loss will incrcasc significantly and thc cfficicncy will dccreasc.
3. Influence of excitation on load characteristics Figure 8. 29 shows the influence of torsional amplitude on load characteristics. It is observed that the torsional amplitude has significant influence on the output characteristics: the no-load speed will arise obviously with the augmentation of torsional amplitude. but the stall torque increases a little. Thc size of thc contact anglc depends on longitudinal amplitude. Fig. 8. 30 indicatcs that the maximal prc-pressurc could incrcase whcn the longitudinal amplitudc increascs and the contact anglc will bc dccrcascd by thc augmentation of thc longitudinal amplitudc under thc samc pre-pressurc. 1600 o;::--~-~--~-~----, ", .
1280 ~
:=
________>:,. . ___ ;__ ----------
U,i~crease://'
320 . ___ [T,-3e-6m
,
960
400r--~-~-~-~-~-.
- - [,,"-le-6m: ----------- [,,"-2e-6m
- - U.- l e-6m U.-2e-61ll ' , : - - - U.- 3e-6m :, _ U.=1 e.-6m :
_. U,-4e-6m
-S
I
0-
640
320
0.08
0. 16
0 2. 4
0.32
Output torque /(N -Ill)
Load characteristics under various U r
Fig. 8. 29
0.4
60
120
180
240
300
360
Contact angle/C)
Fig. 8. 30 Pre-pressure vs. contact angle under various U r
Figure 8. 31 shows thc load characteristics under various longitudinal vibration amplitudes. It is observed that when the longitudinal amplitude is smaller. the load characteristics is soft. namely. the speed will decrease sharply as the augmentation of thc torquc, which is not conductive to thc opcrating.
Howcvcr,
when thc longitudinal amplitudc is larger, thc load charactcristics becomes hard, the spccd will change smoothly as the torquc incrcases. which is favorable for the steady operating. Also. when the longitudinal amplitudc is smallcr. the stall torque and the no-load speed increase obviously by the augmentation of the longitudinal amplitude, but when the longitudinal amplitude is larger, the results are contrary.
Chapter 8
Ultrasonic Motor Using Longitudinal-Torsional···
400
1600
<::;-·:.:.t·:::~··~··
1280
....... : ........... ' '':''
.:
i
8-
- - U, = le-6m ....._....- U, =2c-6m
<"......
~
~
320 ............
"X~. ~ \~
.: ......
...... .
>·\..>t.\'.. . . \
,.
.... \ .. I ..
160
:, i
.:.............
80
:\
\ \ :: \1.
0
0.4
0.16 0.24 0.08 0.32 OlllplIllorqlle f(N ·m)
Fig. 8. 31
,, :': \
240 ~
~ ~
o ~--~--~~--~--~--~
o
. ....... _. ..U,-2e-6m - - - U,-3e-6m (l,=4 e-6m .. : .... \ ... ; ...... .
320
r
........ ....... .....< .. :
640
.... .
~ U, = l e-6m
-- -. _-. UU.-3e-6m =4e-6m
>"~:r:~:,'
960
253
0
60
240
120
\ }""\\. : I
: \ \
.
300
360
OLilplIllorqlle f(N · m)
Load characteristics under various U z
The above analysis illustrates that the augmentation of the torsional amplitude plays an important role in improving the speed of the motor, and the augmentation of the longitudinal amplitude will improve the torque and the efficiency, but when the longitudinal amplitude reaches some value, whose influence on the torque and the efficiency arc small.
4. Influence of friction layer The rigidity of friction layer is influenced by the thickness of friction layer. The rigidity is small when the friction layer is thick, and small pre-pressure will make the contact angle to approach the critical contact angle. In this way, it is difficult to apply larger pre-pressure. The relationship between the pre-pressure and contact angle is shown in Fig. 8. 32 (a), and the corresponding load characteristics arc shown in Fig. 8. 33. Fig. 8. 32(a) indicates that the maximal pre-pressure is 300
1400
250
~
200
i:! Q.
150
~
i!
Cl.
,
.
- - h,=0.3e-3m h f - 0.6e-3m h,=0.9c-3m n
o •• n . o n
---
1120
::::::::~I~.::.>-:.,.~ .. ~"- r::.~I.';~~~~. :
. ~
.,
" ~
........... .. ..... .....
~ II0-
100 50
840
:"'''~~~.
...... ;..... >.:.\ .... "'"
:'\\\
.. ..
f\.
560
- - ' £, - OJ5e-9 ..........• £r- 1.05e-9 \\ 280 . - - - [:'r= 1.7 5e-9 ..:- ............ '(
- . _ . H, =2.45e-9:
0 0
60
120
180
240
300
360
\
O o~--~--~--~: ~--~:0.32--~\0.4 0.08 0.16 0.24 OulPUI lorque !(N ' m)
Conl.cl anglef()
(a) Pre-pressure vs. contact angle under various hi Fig. 8. 32
:
(b) InIluence of various EI on mechanical characteristics
Influence of friction layer
Ultrasonic Motors Technologies and Ap plicalions
254
inversely proportional to the thickness of the friction layer, so the thickness of friction layer should be as thin as possible. Figure 8. 32 (b) shows the load characteristics under various elasticity modulus. It is observed that when the elasticity modulus is larger, the mechanical characteristics is hard, there is an obvious inflection point on the curve and the mechanical characteristics changes smoothly before the point and changes sharply after the point. Although the stall torque and the no-load speed changes slightly, the mechanical characteristics become harder, which is favorable for the steady operating. Figure 8. 33 indicates that the stall torque, no-load speed, and the efficiency will be increased by the decrease of the thickness of the friction layer. However, if the thickness of the friction layer is too thin, it is too difficult to fabricate and easy to separate, which will shorten the life of the motor. 1500
r----~----~----,
60r--~-~------,
- - "r={l·3e-3m
- - ilr ={l.3e-3m 1200
" ~ It
(/)
...........
....•.•..•. " ,={l.6e-3m - - - ilr ={l.ge-3m
il r ~0.6e - 3m
- ",=0.ge-3m
900
600
24 ..
300
12
0,08
0,16
0.24
0.32
0,08
0.4
OutPlillorque /(N ' m)
Fig. 8. 33
0,16 0.24 0.32 Olilput torque /eN 'm)
0.4
Influence of various hi on load characteristics
In order to validate the contact model, the simulation results and the corresponding experimental results arc compared in Fig. 8. 34. 1400
1120
'C '8 840 ";:;
'f& '"
:
:
._..- simulation result
~~1'~"
560
280
::
":
"'T\ "" ~
°0~--O~.O-8---0~,16---0-,2~4---0~,3~2-~-O~.4 OutPlit torque /(N'Ill)
Fig. 8. 34
Mechanical characteristics of mode conversion type ultrasonic motor
Chapter 8
Ultrasonic Motor Using Longitudinal-Torsional···
255
In a word, according to the dynamic simulation analysis, the following conclusions can be drawn: (1) Enhancing pre-pressure is a good way to increase the output torque and contact time and improve the stability, but the no-load speed and efficiency will decrease. So the proper pre-pressure is need to make the motor to operate reliably. (2) Increasing the excitation voltage for torsional vibration will lead to the increase of the rotary speed obviously, while increasing e.Lcitation voltage for longitudinal vibration will improve the output torque and the contact angle, which indicates that they may fit different needs. (3) The elasticity modulus and the fabrication process are important factors for selecting the friction material.
8. 4 8.4. 1
Mode Conversion Type Ultrasonic Motor Structure and Operating Modes
Mode conversion type ultrasonic motor is a kind of LTUM, Refs. [21 J-[ 26 ] present the motor with oblique slots, which has been developed by PDLab, as shown in Fig. 8. 35. The stator part of the motor is composed of mass, PZT pieces for longitudinal vibration, and L-T mode convertor. They are compacted by the fasting bolt to form a sandwiched Langiven vibrator. The rotor part of the motor is composed of rotor, ball bearing, helical spring and lock nuts. The prepressure between the stator and rotor is applied by the helical spring. The stainless steel is used for mass, fasting bolt and rotor, and the duralumin is used for L-T mode convertor. Mass(Steel)
Flange
Fig. 8. 35
PZT for L-mode
Rotor Ball Helical Lock (Steel) bearing spring nut
L-T mode
Driving
Central
convertor
surface
axis
Spring cover
Sketch of mode conversion type LTUM with oblique slots
The L-T mode convertor can transform a part of the longitudinal vibration into the torsional vibration by the oblique slots. Generally, the motor needs only one longitudinal mode to rotate the rotor in unidirection and the mode is called forward direction L-T mode, as shown in Fig. 8. 36(a), but it can carry out reversible rotation at the other modal frequency, and the mode is called opposite direction L-T mode, as shown in Fig. 8. 36(b).
Ultrasonic Motors Technologies and Ap plicalions
256
(a) Forward direction L-T mode
Fig. 8. 36
8. 4. 2
Operating modes of mode conversion type LTUM
Principle of Mode Conversion
There are some primary researches on the principle of the mode conversion using the L-T mode convertor with oblique slots L27 - 28 -. Especially. ] un Pi started from the reflection theory of stress in the oblique slots. and derived the stress state after reflection and the conversion coefficient from input to output stress wave[29:. Figure 8. 37 is the sketch of an L-T mode convertor. which is a hollow pipe with 12 oblique slots that arc cut along its circumference. On the assumption that the mode convertor can be unfolded to a plane. as shown in Fig. 8. 38. the surfaces of oblique slots can be regarded as free surface.
Oblique slot
Fig. 8. 37 Sketch of L-T mode convertor with oblique slots
Fig. 8. 38 Longitudinal wave acting on oblique slot
From stress wave theory:30 91: and the analytical method proposed by ] un Pi. we consider that the reflection longitudinal wave P 2 and the reflection shear wave SV are produced simultaneously when the incident longitudinal wave P 1 reaches the free surface of oblique slots. Since every two slots are parallel and the interval distance of them is much less than the wavelength. the incident longitudinal wave P 1 • the reflection longitudinal wave P 2and the reflection shear wave SV can generate a superposition in the slots. The Stress wave after superposition can be decomposed to axial and circumferential components. which will produce corresponding vibrations on the end surface of the mode convertor. It is assumed that all the longitudinal wave can be reflected. the reflection wave ineludes the longitudinal wave and the shear wave. a expresses the incident angle of the longitudinal wave P 1 and the reflection angle of the longitudinal wave P2
•
f3
expresses the reflection angle of the shear wave SV. (90° - a) expresses
Chapter 8
Ultrasonic Motor Using Longitudinal-Torsional···
257
the angle between the oblique slots and the axis of the mode convertor. Taking micro segmcnt from thc surface of the oblique slots and considcring thc strcss states induccd by thc incident longitudinal wavc P r
,
the reflection longitudinal
wave P, and the shear wave SV are shown in Fig. 8. 38. It is assumed that the incidcnt longitudinal wavc intcnsity and thc reflcction longitudinal wavc intcnsity are ar and a, • respectively. the shear wave intensity is (a), (b), and (c).
(a) Stress slate induced by P I
Fig. 8. 39
tress state induced by P2
(b)
T,.
as shown in Fig. 8. 39
(c) Stress slale induced by SV
Stress states induced by incident and reflection wave
From sound wave reflection principle, we have slna sinf3
(8. 33)
That is ~ _ 1- /1 -
sin' a - 2sin' f3 sin'a
(8. 34)
wherc /1 is the Poisson's ratio. C L is the velocity of longitudinal wave, C r is thc velocity of shear wave. From the equilibrium conditions on thc surfacc of the slot, we can obtain: {
a, cos2f3 -
T,
sin2f3
a, sin2f3 tanf3 cota
= -
ar
cos2f3
+ T2 cos2f3 =
ar
sin2f3 tanf3 cota
(8. 35)
Then thc longitudinal wavc strcss and thc shcar wavc strcss can bc expressed as a, {
T,
arR -
R=
ar (1
+ R) cot2f3
tan'2ptanp- tana tan' 2f3 tanf3 + tan a
(8. 36)
The matcrial of thc mode convertor is duralumin, whosc Poisson's ratio
o. 33.
IS
According to the Eq. (8. 36), stress curves of two reflection waves can be
obtaincd, as shown in Fig. 8. 40. Abovc analysis indicatcs that the mode conver-
tor can transform the part of longitudinal vibration to the torsional vibration.
Ultrasonic Motors Technologies and Ap plicalions
258
-0.2
r--~--~-~--~---'
0.6 048
-0.52 ...... .
0.36
f
b
~
... N
-0.68 .......... .
0.24
-0.84 -l~~~--~-~-~~-~
o
18
36 54 Incident angle a (a)
Fig. 8. 40
8. 4. 3
72
90 Incident angle a (b)
"l. and .:r:z... vs. incident angle a 0'1
0'1
Design of Mode Conversion Type L TUM with Holes
Since the processing technique of oblique slots is difficult. a new mode conversion type of L TUM with holes is developed by PDLabL32-33-. which uses the ordered arrangement of holes to realize the function of oblique slots. as shown in Fig. 8. 41. This structure can simplify the processing technique and reduce cost. In order to obtain good characteristics. the longitudinal and torsional amplitude must be optimized. Since the longitudinal amplitude supports the output torque of the motor and the torsional amplitude controls the speed of the motor. the longitudinal and torsional amplitudes should be maximal simultaneously in the design. Based on the ANSYS analysis. the structure parameters of the motor can be determined. Mass(Steel) PZT for L-mode
L-T mode convertor with holes
Ball
Driving surface
Helical Lock
Central axis
Sketch of mode conversion type LTUM with holes
Fig. 8. 41
Fig. 8. 42 Structure of the stator
The structural parameters of the mode conversion type LTUM with holes are shown in Fig. 8. 12. The initial parameters are shown in Table 8. 3. by which the FEM model can be established. When applied 100Vpp voltage to the PZT pieces and each parameter perturbs by 10 % of itself. the relative sensitivity of longitudinal amplitude and torsional amplitude can be solved. as shown in table 8. 3.
Chapter 8
Ultrasonic Motor Using Longitudinal-Torsional···
Table 8.3 parameter
Value(mm)
259
Initial values of structural parameters of stator
PI
Pz
P,
P4
Ps
P6
P7
Ps
Po
5
5
23
4
16
2
2.5
( 45')
37
Sensitivity of torsional vibration
o.
084
0.074
0.242
0.008
Sensitivity of longitudinal vibration
o.
082
o.
0.818
-0.066
122
-0.056 -0.466 -0.088 -0.399 0.342 1
-0.508 -0.088 -0.021 0.316
The analysis of A:'\ISYS indicates that the parameters P, , P s , P e , PH and P g have greater influence on longitudinal and torsional amplitude, thus we can adjust these parameters to obtain suitable longitudinal and torsional amplitude.
8. 4. 4
Testing
1. Load characteristics under various friction pairs The friction layer can directly influence the motor performance and life. Hence it is important to choose suitable friction pairs. Friction behavior and wear loss depend on all components participated in friction and various loads L31 -. The friction pair is defined as the combination of friction material on the stator and friction material on the rotor. Applied 10 friction pairs in Table 8. 1 to the new mode conversion type LTUM with holes, as shown in Fig. 8. 13, of which the diameter is 15mm. The testing results indicate that different types of friction pairs have obvious influence on the load characteristics. Fig. 8. 44 is the load characteristics using :'\10. 3 (soft/hard) friction pair and Fig. 8. 15 is the load characteristics using :'\10. 10 (hard/soft) friction pair. in which the load characteristics is the best by using hard/soft friction pair. the maximum revolution. stall torque, and efficiency are 1 082r/ min. 0.32:'\1 ·m, and 32.3 % , respectively. In Figs. 8. 44 and 8.45, P io ' P n, and r; indicate input power, output power, no-load speed, and Output efficiency. OUI
Fig. 8. 43
'
Mode conversion type LTUM with holes
It is observed that when the hard/soft friction pair is adopted, the load characteristics is the best, the better is the hard / hard friction pair, and the
260
Ultrasonic Motors Technologies and Ap plicalions
worst is the soft / hard friction pair. When the same friction pair is used, the load characteristics of the motor with teeth on stator are better than that of the motor without tooth on stator. The motor with teeth on stator runs stably. and has better load characteristics, and the wear loss is less than that of the motor without teeth on stator. Table 8. 4 Materials of
Materials of
stator surface
rotor surface
Aluminum
Friction Pairs used for testing No-load speed/Cr/min)
"fo.
Type
Ceramic
1
Soft/hard
822
Aluminum Ctooth)
Ceramic
2
Soft/hard
Resin- based material
Ceramic
3
Aluminum
Stainless steel
Aluminum Ctooth)
Stall Output torque/CN-m) efficiency / 0.113
15.0
1 181
o.
160
13. 6
Soft/hard
517
o.
108
4.0
1
Soft/hard
872
o.
161
12. 1
Stainless steel
5
Soft/hard
1 325
o.
181
18.5
Ceramic
Stainless steel
6
Hard/hard
1 104
0.247
19.8
Ceramic
Ceramic
7
Hard/hard
852
0.203
10.9
Ceramic
Resin-based material
8
Hard/ soft
828
0.241
23.9
Aluminum
Resin-based material
9
Hard/ soft
1 019
0.265
27.3
Aluminum Ctooth)
Resin-based material
10
Hard/ soft
1 082
o.
32.3
320
%
Chapter 8
10
Ultrasonic Motor Using Longitudinal-Torsional···
8
40
80
6
30
60
~
~.<
o!
•••
Pm
96
.s
~
E
~
::-
20
40
2
10
20
o
o
o
0.40 Output lorque /(
Fig. 8. 45
n 'I
-.:::
~
"-
1200 , -------------------------------- ,
100
50
261
'm)
Measuring load characteristics using :'\[0. 10 friction pair
2. Load characteristics with various pre-pressure springs When different springs arc used for the new mode conversion type L TUM, the load characteristics change greatly. Table 8. 5 shows 1 kinds of springs used for providing pre-pressure and the summary of the load characteristics of the motor, of which the diameter is 15mm. Figs. 8.46 and 8. 47 show the load characteristics of the motor by using No.1 and )Jo. 4 springs. It is observed that the load characteristic using non-linear spring is the best. Table 8. 5 :'-10.
Types of spring
1
compression spring
Four pre-pressure springs used for testing
Features
Lcngth /mm
:'-la-load Output Diameter Stall spccd(r/min) torquc/(N·m) efficicncy / /mm
Cylindrical helical Linear
15
10
955
O. 200
12.2
Linear
15
10
1 125
O. 254
19.4
20
top 7. 1 bottom 9. 2
1 002
0.292
18. 1
4
9
982
O. 315
21. 2
(d= 1. 2)
Cylindrical helical 2
compression spring
(d= 1. 1)
3
4
Conical coil spring
Silicon rubber flat spring
Wcak nan-
lincar Strong non-
lincar
%
262
Ultrasonic Motors Technologies and Ap plicalions
50
8,0
1000
100
o 6.4
40
80
o
o
800
... D OD
4,8
~
•
"-
~ "-
~ 20
"'" 40
1.6
10
20
0,0
a
0
3,2
P in
60
30
" 1/
" ~ -.
'"
400
200
0,05
0.10
0,15
0.20
0.25
Olllput torqu e /eN ' m)
Fig. 8. 46
Measuring load characteristics using :'\[0. 1 spring
In a word. the stator with teeth. hard/soft friction pair and non-linear spring can significantly improve the load characteristics of L TUM. 8.0
50
100
6.4
40
80
4.8
30
60
1000 '/
~g
"-
~
3.2
"-
'# 20
<::
40
1.6
10
20
00
o
o
P ill
,....,
':§,"" '"
0.40 OUlplIl lorqlle /(N ' ilI )
Fig. 8. 47
Measuring load characteristics using No.4 spring
References [ 1]
Chunsheng Zhao. Ultrasonic motor tcchniques in the 21st ccntury. ] ournal of Vibration
Chapter 8
[ 2J
[ 3J
[ 4J [ 5J [ 6J
Ultrasonic Motor Using Longitudinal-Torsional···
263
Measurement & Diagnosis, 2000, 200): 7-12. (in Chinese) Heming Sun, Chunsheng Zhao, Xiaodong Zhu. Recent advances in ultrasonic motor using LT vibration modes. Journal of Vibration Measurement & Diagnosis, 2002, 22 (1): 9-11. (in Chinese) K Nakamura, M Kurosawa, S U eha. Characteristics of a hybrid transducer-type ultrasonic motor. IEEE, Transactions on Ultrasonics Ferroelectrics and Frequency Control, 1991, 38 (3): 189-193. K "fakamura, S Ueha. Performances of a hybrid transducer type ultrasonic motor as a function of the size. IEEE Ultrasonics Symposium Proceedings, 1991, 1-3: 557-560. J Satonobu, "f Torh, K "fakamura, et al. Construction of megatorque hybrid transducer type ultrasonic motor. Japanese] ournal of Applied Physics, 1996, 35 (9B): 5038-5011. Y Tomikawa, K Adachi, M Aoyagi. Some constructions and characteristics of rod type piezoelectric ultrasonic mOlor using L- T vibrations. IEEE, Transactions on [lltrasonics, Ferroe-
lectricsand Frequency Control, 1992, 39(5): 600-608.
[ 7J
[ 8J [ 9J [10J [11]
[12J [13J [11J [15J [16J [17J [18J [19J [20J [21J [22J
[23J [24J
Hengbing Zhao, Xiaodong Liu, Yongxiao Chen, et at. Analysis of the structure of ultrasonic motor with large torque. Small & Special Electrical Machines, 1999, 27(4): 15-18. (in Chinese) Heming Sun, Chunsheng Zhao. Structure optimization of ultrasonic motor using L-T mode. Micromotors Servo Technique, 2002, 35(4): 7-10. (in Chinese) Heming Sun, Chunsheng Zhao, Xiaodong Zhu. Experimental study of ultrasonic motor using L-T mode. Small & Special Electrical Machines, 2002, 30 (4): 3-8. (in Chinese) Zheng Tao, Chunsheng Zhao. Brush type ultrasonic motor using L- T vibration modes. Chinese Invention Patent, 200110011953, 2001-05-20. Zheng Tao. Research on Ultrasonic Motor Using L- T Vibrator. Dissertation for the Degree of Doctor of Philosophy. "fanjing: "fanjing University of Aeronautics and Astronautics, 2006. (in Chinese) J van Randeraat, R E Setterington. Piezoelectric Ceramics. Beijing: Science Press, 1981. (in Chinese) B Jaffe. Piezoelectric Ceramics. Beijing: Science Press, 1979. (in Chinese) Tetsuro Tanaka. Piezoelectric Ceramic Materials. Beijing: Science Press, 1982. (in Chinese) Daoren Song, Mingshan Xiao. Piezoelectric Effect and Applications. Beijing: Popular Science Press, 1987. (in Chinese) Jinfcng Wang, Zutong Jiang, Ruida Shi. Piezoelectric Vibration. Beijing: Science Press, 1989. (in Chinese) Fuxue Zhang, Liqun Wang. Modern Piezoelectricity. Beijing: Science Press, 2002. (in Chinese) Wei Feng, Xizhong Wu, Yanding Wei, et at. Principle and research on the modal matching of the L-T composite USM. Micromotors Servo Technique, 2001, 31(6): 7-12. (in Chinese) Zhongmao Lin. Principle and Design of Ultrasonic Horn. Beijing: Science Press, 1987. (in Chinese) S Ueha, Y Tomikawa. Ultrasonic Motors Theory and Applications. Oxford: Oxford Science Publications, 1993. Jifeng Guo, Shujuan Gong, Xiao Liu, et at. The study of characteristics on the L-T ultrasonic motor. Acta Acustica, 2003, 29 (4): 334-340. (in Chinese) Jifeng Guo, Yanding Wei, Xiao Liu, et at. Force transfering model of hybrid transducer type ultrasonic motor. Proceedings of the Chinese Society for Electrical Engineering, 2003, 23 (5): 80-85. D Karnopp. Computer simulation of stick-slip friction in mechanical dynamic systems. ] ournalof Dynamics Systems, Measurement, and Control, 1985,107(1):100-103. J Tsujino, It Suzuki. Load characteristics of ultrasonic motor with a longitudinal-torsional converter and various nonlinear springs [or inducing static pressure. IEEE Ultrasunics SY7n-
posium. Atlanta, GA: IEEE, 2001: 515-550.
264
Ultrasonic Motors Technologies and Ap plicalions
[25J
1 Tsujino, R Suzuki, H Yasojima. Load characteristics of ultrasonic rotary motor using a L-T
[26J
vibration converter. IEEE Ultrasonics Symposium. San Antonio, TX: IEEE, 1996: 377-382. J Tsujino, T Uchida, K Yamano. Ultrasonic plastic welding using two 27 kHz complex
[27J [28J [29J
vibration systems. IEEE Ultrasonics Symposium. Toronto, Ont. : IEEE, 1997: 855-859. Y Koike, F Magane, K Nakamura, et al. A vibration analysis of a L-T coupling vibrator with oblique slots. 1997 World Congress on Ultrasonics. Yokohama, 1997. Shuyu Lin. Study on the L-T composite transducer with slanting slots. Acta Acustica ,1999, 24(1): 59-65. (in Chinese) J un Pi. L-T vibration converter of cylinder with multiple diagonal slits. Chinese] ournal of Mechanical Engineering, 2008, 11(05): 212-218. (in Chinese)
[30J
Xiaoqing Ma. Impact Dynamics. Beijing: Beijing Institute of Technology Press, 1992. (in Chinese)
[31J
Hongwcn Liu. Mechanics of Materials (on list). Beijing: Higher Education Press, 1992. (in Chinese)
[32J
Lin Yang, Jiamci Jin, Chunshcng Zhao. An ultrasonic rotary motor by using longitudinaltorsional vibration converter with holes. Journal of Vibration, Measurement & Diagnosis, 2009(2): 133-136. (In Chinese)
[33J
[34J
Lin Yang, Qingjun Ding, Chunsheng Zhao, et al. Load characteristics of ultrasonic motor using hongitudinal-torsional convertor with diagonal slits under various friction pairs. Proceedings of the CSEE, 2010(15): 91-98. (In Chinese) lunli Shi. Research on Mathematical Driving Model of Ultrasonic Motor and Its Frictional materials. Chengdu: Southwest liaotong University, 2004. (in Chinese)
Chapter 9
Linear Ultrasonic Motors As a type of ultrasonic motors, a linear ultrasonic motor (L USM) also utilizes the converse piezoelectric effect of piezoelectric ceramics and the ultrasonic vibration of an elastic body:1:. It transfers the micro amplitude motion of a stator into the macro linear motion of a slider by the friction force between the stator and slider. Besides the common characteristics of rotary ultrasonic motors, linear ultrasonic motors also possess the following eharaeteristies: 2:
:
(a) the capability of
a direct and straight drive; (b) a high precision accuracy up to nanometer level; (c) good control characteristics due to no movement errors from auxiliary parts, such as linkage, ball screw, transmission belt, etc; (d) a simple structure and the variability of shape, allowing easy miniaturization and weight loss. Linear ultrasonic motors have been developing rapidly in recent years. Their applications are more and more widely used in the following areas: (1) semiconductor manufacture devices;
(2) aeronautic and astronautic appearances; (3) precise position stages; (4) biomedical equipment; (5) optic fiber alignment facilities; (6) miniaturization of information systems. To date, there arc several methods to elassify linear ultrasonic motors. Based on the form of wave motion, they can be elassified as standing wave and traveling wave types. Hereinto, according to the number of a stator's driving feet, standing wave type can be elassified into single foot and multi-foot types, and traveling wave type can be elassified into straight beam and ring beam types. Based on the way of the elliptical motion generated, they can be elassified into single mode and multi-mode types. Based on the direction of displacements of a stator, they can be elassified into in-/out-of-plane vibration types. Based on the relationship of relative motion, whether the piezoelectric vibrator used as slider or stator can lead to different elassifieations, either as self-moving (the piezoelectric vibrator is used as a slider) or non-self-moving (the piezoelectric vibrator is used as a stator). Figure 9. 1 shows the elassifieation of a linear ultrasonic motor based on the description aforementioned.
C. Zhao, Ultrasonic Motors © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011
Ultrasonic Motors Technologies and Ap plicalions
266
Classification oflinear ultrasonic motor
I
According to wave- I I According to morion of I mO\1on ronn piezoelectric vibrator ,-L-
-'-
">-
rg;
I
By the way of ell iptical motion generat ion
r-
II
,-L-
According to I displacement direction I -'-
-'-
";:
'"~
;!>
.S <;
.~ 0
l-
V)
.r " <;
e>-
~
0
E
<..!.
~
is Z
'--
"0
.",
E
" ~
in
'----
'"0
.",
~ ~
'----
~
ii
is.
""
<..!.
~
6
-
'" ~ .==
-
E
"'"
.&>
00
~
E..,
= Vi ii: - '§
Fig. 9. 1
9. 1
Classification of linear ultrasonic motors
State of the Art of Linear Ultrasonic Motors
In 1985, based on the study of traveling wave type rotary ultrasonic motors, Sashida presented two types of traveling wave linear ultrasonic motors: straight beam and ring beam types:':. Fig. 9. 2 shows the prototype of a straight beam type linear ultrasonic motor. In order to produce the traveling wave, two Langevin vibrators were fixed on both ends of the beam, respectively. One was used as a vibrator, and the other was used as an absorber to prevent the reflection of the traveling wave (see Section 4.3.4). The traveling wave on the beam was produced to drive the slider when Langevin vibrators were connected to power source. The slider would move in an opposite direction when the role of two vibrators was exchanged. In 1998, Hermann, Schinkothe, et a1. developed the prototype of ring beam type linear ultrasonic motor, as shown in Fig. 9. 3. To prevent the reflection of wave, a elose ring beam was chosen as the stator, and the traveling wave on the surface of the ring beam was excited by the piezoelectric ceramic pieces bonded inside of the ring beam. The slider would perform the linear motion when it was pressed on the linear portion of the ring beam. Slider
Fig. 9. 2
Pre-pres ure
Straight beam type
linear ultrasonic motor
Fig. 9. 3
Ring beam type linear
ultrasonic motor
Chapter 9
Linear Ultrasonic Motors
267
In 1997, the company in Japan produced the prototype of a thin linear ultrasonic motor-' - , as shown in Fig. 9. 4. It utilized two triangle piezoelectric ceramic pieces as the vibrator. Its maximum no-load speed was 4S0mm/s, and the maximum thrust was 3N, and the highest efficiency was 10 %. Subsequently, PI Company in Germany also developed similar linear ultrasonic motor L5J , as shown in Fig. 9. S. The no-load speed of the motor was up to 600mm/s, and the maximum thrust was 60)J.
Fig. 9. 4
Thin linear ultrasonic motor
Fig. 9. 5
PI linear ultrasonic motor
In 1998, Wakai, Kurosawa, et al. developed the transducer for linear ultrasonic motor:':, as shown in Fig. 9. 6. The speed without load of the motor was 3. 5m/s, and the maximum thrust was SIN. Mountin g block
Rear-end block
Driving foot
Fig. 9. 6
Transducer for linear ultrasonic motor
Recently, many kinds of linear ultrasonic motors have been developed in all the world. In 1999, the linear ultrasonic motor, studied by Kyocera Company in Japan[7] , was put into use to drive directly x-y precision stages in vacuum environment. This precision stage was developed for the chemistry analysis, thickness measurement, and large scale integrated circuit (LSI) manufacture. In 2001, Anorad company developed series of multi-head linear ultrasonic motors (PCLM):8: , as shown in Fig. 9. 7. It could realize step movement with nanometer position accuracy. The maximum speed was 2S0mm/s, the maximum thrust was from S. 3)J to 21. 5N through different series of motors. A tiny Squiggle linear ultrasonic motor developed by New Scale Technology in USA is shown in Fig. 9. 8: 9: . Its operating principle is similar to the bar-type ul-
268
Ultrasonic Motors Technologies and Ap plicalions
trasonic motor described in Chap. 7. The motor consists of a nut with piezoelectric ceramic pieces bonded on its surface and a screw bolt matched with the nut, and utilizes the first orthogonal bending modes of the nut. The elliptical motion in the circumferential direction of the points on the inner thread was generated. The screw bolt would be driven linearly while the nut was fixed. The speed of SQ-100-)J series of motors ranged from 1fLm/s to 10mm/s, the maximum thrust was up to S)J, and the highest position accuracy was up to 20nm. It has shown potential application prospect in the portable products, such as the micro cameras and wearable medical devices.
Fig. 9. 7 PCLM developed by Anorad company
Fig. 9. 8 Squiggle linear ultrasonic motor
Figure. 9. 9 shows the impact type piezoelectric motors based on the inertial prineiple- 1o - ll - , which can be used in the varifocal systems of micro cameras.
(a) Prototype
Fig. 9. 9
(b) Application in micro-camera
Inertial impact type linear piezoelectric motor
In China, researchers from numerous universities and scientific research facilities have studied and developed linear ultrasonic motors. In 1997, a standing wave type linear ultrasonic motor with double driving feet developed by Weidong Liu was applied to the servo feedback system of a tenuous spark machini ng c12:. In 1998, the self-moving type linear ultrasonic motor with" 1(" shaped vibrator was studied by Cheng lin GU_ 13 _. In 2001, Chaodong Li, Chunsheng Zhao, et al. developed a mini and bionic linear ultrasonic step motor Ll1 - , which was driven by
Chapter 9
Linear Ultrasonic Motors
269
a single-phase power source, and possesses convenience for wireless control and capability of perpendicularity walking without any assist. The maximum speed was up to 80mm/s, and the maximum inertia load achieved was lS:'\I. In 1996, PDLab succeeded in developing a straight beam type traveling wave linear ultrasonic motor15-16J, as shown in Fig. 9. 10 Ca). In 1998. Chunsheng Zhao. Chaodong Li, et al. developed the self-moving linear ultrasonic motor with large thrust based on longitudinal and bending vibration modcs: 1? 18:, as shown in Fig. 9. 10Cb). The maximum speed was up to 510mm/s, and the maximum thrust was up to llN. In 2001. Chunshcng Zhao and Jian Liu developed the linear ultrasonic motor based on the in-plane modes[1,19 21: , as shown in Fig. 9. 11. The speed of this motor was SOmm/ s, and the maximum thrust was up to 5)J while it was driven by an AC signal with a frequency of 42. 8kHz and amplitude of 2S0V pp •
Straight beam type linear ultrasonic motor
Fig. 9. 10
Linear ultrasonic motor based on modes in-plane
Fig. 9. 11
In 2003, two types of linear motion stages using linear ultrasonic motors as actuators were developed in PDLab- 22 - 21J . One stage had the stroke of 100mm. maximum speed of 60mm/ s and maximum thrust of 23)J. The other one also had the stroke of 100mm, maximum speed of 6Smm/s, and maximum thrust of 50:'\1. In 2006, Chunsheng Zhao, Jiamei Jin, et al. presented the new type of linear ultrasonic motor based on the out-of-planc modcs L25J , as shown in Fig. 9. 12. The motor utilized the two orthogonal second bending modes of a square thin plate. The teeth on the plate were arranged in a way to produce elliptical motion with different phases. Thus the teeth can drive the slider pressed on it alternately during every vibration cyele. Under a pre-pressure of 80)J, the maximum speed of the motor was 180mm/ s, and the maximum thrust was S)J, while it was driven by an AC signal with a frequency of 57. 36kHz and amplitude of 200Vpp • In 2008. Chunsheng Zhao, Zhiyuan Yao, et al. designed the V shaped linear ultrasonic motor with double horns[25:, as shown in Fig. 9. 13. The maximum speed of the motor was 235mm/ s and the maximum thrust was 21. 1:'\1. Recently. the two type stages with two degrees of freedom driven by linear ultrasonic motors have been developed by PDLab, as shown in Fig. 9. 14. In the chapter. the operating principle and design method of linear ultrasonic motors arc illustrated by several examples of linear ultrasonic motors developed by PDLab.
Ultrasonic Motors Technologies and Ap plicalions
270
Fig. 9. 12 Linear ultrasonic motor based on out-of-plane modes
Fig. 9. 14
9.2
Fig. 9. 13 V shaped linear ultrasonic motor
X- Y linear stage
Linear Ultrasonic Motors Based on d31 Effect
As mentioned in Chap. 2, piezoelectric ceramic elements have four vibration types. LE type is the extension-contraction mode along the length direction perpendicular to the electric field direction, which utilizes the d'l effect of piezoelectric ceramic. Moreover, the in-plane mode type linear ultrasonic motors often use the PZT's d'l effect, which can simplify the structure of the motor, and make it easy to realize miniaturization.
9. 2. 1
Linear Ultrasonic Motor with Double Driving Feet
1. Motion mechanism In this section, the in-plane modes of a thin rectangular plate arc analyzed and how to realize the elliptical motion on the driving foot of the stator is described: (1) Vibration of a rectangular thin plate As shown in Fig. 9. 15(a). the stator is simplified as a rectangular thin plate. where a, land b is the thickness, length, and width of the simplified plate, respectively. The first longitudinal in-plane mode El (Fig. 9. 15 (b» and the second bending in-plane mode B, of the plate (Fig. 9. 15(e» arc chosen as the oper-
Chapter 9
271
Linear Ultrasonic Motors
a ting modes of the linear ultrasonic motors.
{¢Ee X)
=
cos(7·r
)
¢B, (x)
=
cosh.B2 X
+
G, (sin.B2 x
cosfh x -
+
(9. 1)
sinh.B, x)
where. the symbols of the equation refer to Chap. 1.
!
~(\V)
I1 I1 111111 1 11 1111 111 11 '
b
(b) First l ongitudinal mode s hape
0
X( I /)
I
(e) Second bending modeshape
(a) Simplified tator
Fig. 9. 15
Simplified stator of linear motor and its mode shapes
To e.rcite these two in-plane operating modes at the same time with the same driving frequency. the plate must be designed reasonably to make the two operating mode frequencies as close as possible. According to Chap. 4, it's hard to obtain the analytical rcsults of the platc under free-frec boundary condition. Therefore, the natural frequency of the plate is often calculated by using finite element method. (2) Motion mechanism Figurc 9. 16(b) shows the structurc of thc stator. Figs. 9. 16(a) and (c) arc the displacement mode shapes and strain mode shapes of E] and B2 , respectively.
z(w)
k~ (a) Displacement mode shapes
o
x(u)
cosmt
sinmt
(b) Stator structure
cosmt
~ B2
(c) Strain mode shapes
Fig. 9. 16
Stator structure and its displacement mode shapes and strain mode shapes
The stator is a rectangular thin plate, which has two driving feet located in the wave crest (x= O. 3l6Z) and trough (x= O. 681Z) of the second bending mode of
272
Ultrasonic Motors Technologies and Ap plicalions
the thin plate. As described in Chap. 1, the piezoelectric ceramic pieces should be bonded to the location of the maximum strain points of the mode. According to the strain mode shapes as shown in Fig. 9. 16 (c), the two piezoelectric ceramic pieces with the same dimension and opposite polarization direction are bonded to the side of the stator. Another piezoelectric ceramic piece polarized in throughthickness direction is bonded to the center of the surface of the stator. as shown in Fig. 9. 16(b). If the effect of the disturbing modes and damping is neglected, the first longitudinal mode and the second bending mode of the stator can be excited at the same time when the stater is driven by two special AC voltages with a 90 phase difference, as shown in Fig. 9. 16(a), the displacement response of the stator can be described as 0
{
=
w(x, t) =
u(.r. t)
(9. 2)
In fact, there is also a phase difference of rr/2 in space between the two displacement responses because the displacement directions arc along the x and z axes, respectively, which then can realize the elliptical motion on the tip of the two driving feet. The motion of the driving feet in a cyele is analyzed in detail, as shown in Fig. 9. 17. In consideration of elear expression, the linear ultrasonic motor designed is supposed as self-moving type. namely, the piezoelectric vibrator acts as a mover. The operating states of the motor are described as follows.
~.
(
Fig. 9. 17
.0.
i
Opcrating principle of the linear ultrasonic motor
(a) Whenwt = 0 ~ rr/2, w(O. 316l,t) =-1 ~ 0, w(O. 684l,t) = 1 ~ 0, u(O. 316l,t) = 0 ~
Chapter 9
Linear Ultrasonic Motors
273
longitudinal mode. Meanwhile, the right foot moves clockwise from the wave crest position of the bending mode down to the largest stretch position of the longitudinal mode. During this time, the left driving foot pushes the mover half step rightward and the right driving foot separates from the rail. (b) Whenwt = rr/2 ~ rr, w(O. 316l,t) = 0 ~ 1, w(O. 681l,t) = 0 ~-1, u(O. 316l,t) = ¢E (0. 316l) ~ 0, u(O. 684l,t) = ¢E, (0. 684l) ~ 0, the vibration state of the mover gradually changes from state (2) to state (3). In the course of changing, the left foot moves clockwise from the largest stretch position of the longitudinal mode up to the wave crest position of the bending mode. Meanwhile, the right foot moves clockwise from the largest stretch position of the longitudinal mode down to the trough position of the bending mode. During this time, the left foot separates from the rail and the right foot pushes the mover half step rightward. (c) Whenwt = rr ~ 3rr/2, w(O. 316l,t) = 1 ~ 0, w(O. 684l,t) =-1 ~ 0, u(O. 316l, t) = 0 ~ -¢E (0. 316l), u(O. 681l,t) = 0 ~ -¢E, (0. 681l) , the vibration state of the mover gradually changes from state ® to state @. In the course of changing, the left foot moves clockwise from the wave crest position of the bending vibration down to the largest shrink position of the longitudinal mode. Meanwhile, the right foot moves clockwise from the trough position of the bending mode up to the largest shrink position of the longitudinal mode. During this time, the left foot separates from the rail and the right foot pushes the mover half step rightward. (d) Whenwt = 3rr/2 ~ 2rr, w(O. 316l,t) = 0 ~ -1, w(O. 684l,t) = 0 ~ 1, u(O. 316l,t) =-¢E, (0. 316l) ~ 0, u(O. 681l,t) =-¢E, (0. 681l) ~ 0, the vibration state of the mover gradually changes from state @ to state CD. In the course of changing, the left foot moves clockwise from the largest shrink position of the longitudinal mode down to the trough of the bending mode. Meanwhile, the right foot moves clockwise from the largest shrink position of the longitudinal mode up to the wave crest position of the bending mode. During this time, the left foot pushes the mover half step rightward and the right driving foot separates from the rail. The analysis aforementioned shows that both the left and the right feet of the mover push independently the mover one step in one cycle. :'\lamely, the stator pushes the mover two steps rightward in one cycle. This case is just like a running leopard (sec Fig. 9. 17). If the phase difference in time between the exciting voltage is changed, such as the sinwt becoming - sinwt, the direction of the elliptical motion will be reversed, and the motion direction of the mover will be reversed as well.
2. Optimal design of the stator This motor mainly consists of two parts, the stator and the mover. The commercially available slider can be used as the mover. The key of design lies in the stator. The designs of the stator include the choice of its operating modes, materials, structure dimension parameters and so on. In general, as to linear ultrasonic motors based on in-plane modes, several factors should be considered as follows:
Ultrasonic Motors Technologies and Ap plicalions
274
(a) The two operating mode frequencies of the stator should be in the ultrasonic frequency range, and as close as possible. (b) The two operating modes should have an overlapped nodal plane used for clamping the stator to reduce the influence of clamping to the operating modes. (c) Driving feet should be located at the wave crest and trough position of the bending mode. (d) The piezoelectric ceramics should be bonded at the maximum strain points of the operating modes. Here, the first longitudinal mode E t and the second bending mode B, arc chosen as the operating modes of the motor, the phosphor-bronze is selected as the material of the stator, and PZT8 is chosen as the piezocleetrie material. The optimal target of the motor is to minimize the natural frequency difference between the two operating mode frequencies by adjusting the structure parameters of the stator. To get design results quickly, effectively and accurately, the optimizing design modular of the finite element software is adopted. Figure 9. 18 shows the stator's structure parameters. The parts with many dots in this figure are the piezoelectric ceramic pieces. The original structure dimension of the stator is shown in Table 9.1. Fig. 9. 19 is the finite clement modcl of the stator. The optimization process of the stator is as follows:
_ .ri..U,",-, Elastic body
Piezoelectric ceramics
Fig. 9. 18
Fig. 9. 19
Sketch of the stator structure parameters
Table 9. 1
Initial value of the stator structure parameters (Unit: mm)
Parameters
Values
Finite clement model of the stator
b 40
8
I,
a 15
4
13
I,
11
(1) Selection of the design variables In order to solve the problem of the consistency of the two operating mode frequencies. the stator's structure parameters which have much influence on the op-
erating frequencies are chosen firstly by sensitivity analysis. From Fig. 9. 18. the whole stator has ten structure parameters. among which the sizes of l, It , l, • a. a t , and b2 are fixed, that is, the length and thickness of the stator are fixed, the
driving feet arc set to the wave crest and trough of the second bending mode, and
Chapter 9
Linear Ultrasonic Motors
275
the width and thickness of the piezoelectric ceramic pieces are also fixed. Therefore. only four structure parameters l2 , l3 • b. and b l are selected to carry out the sensitivity analysis. The corresponding expression of the sensitivity is
{
s
= bJ
S
aib CJp;
ibm - i
=
11m -
CJf,
ap;
I;
IfJ
Lp;
Lp;
(9. 3)
fto
ib is the second bending mode frequency of the stator; i, is the first longiib and ibm are the second bending mode
where.
tudinal mode frequency of the stator.
0
frequencies of the initial structure and the modified structure of the stator, respectively.
i,o
and
i'm
are the first longitudinal mode frequency of the initial
structure and the modified structure of the stator, respectively. Lp; is the small change of the stator's structure parameter Pl' The value of Lp; is set to be O. OOlm. According to Eq. (9.3), the sensitivity analysis of the operating mode frequencies to structure parameters is carried out by using the finite element model shown in Fig. 9. 19. The analysis results are shown in Figs. 9. 20 and 9. 21. The parameter of pj (j
=
1 , 2 ,3 ,1) is correspond-
ing to the structure parameters of l2' l3 , b, and b1 •
1500
.,
E E
1000
;.,
J:
500
~
0
.,
;.,
"'" :~
2000
.;;;
-500
Of)
-1 000
5
E E
J:
"'"
~
-500 .' .," .......... ;...........i ..... _..... ;........... ;........ . -600 .'
'~
.~
, ..--"
2
;;
....-.....
PI
- 700 / -800
Of)
3
4
Fig. 9. 20
./" 2
3
4
Fig. 9. 21
Scnsitivity of bending mode frequency to structure parameters
Sensitivity of longitudinal mode frequency to structure parameters
According to the results of the sensitivity analysis above. three structure parameters are choscn as dcsign variables. )J amely,
x
=
[l2
b
(9. 4)
blJ
(2) Objectivc function The frequency function of the two operating modes is described as
iB, H cncc, thc
0
=
iB ex),
iE,
=
iE ex)
(9. 5)
bj ccti ve function of thc optimal dcsign is Fob;
=
min
liB, - iE, I
(9. 6)
(3) Optimization procedurc and results In this section, APDL of the ANSYS, a parameter programmablc language,
IS
276
Ultrasonic Motors Technologies and Ap plicalions
adopted to realize the whole optimization design. Fig. 9. 22 shows the flow of the optimization design. Optimization Analysis
r-,-_- _-_- -_ _-_- _-_- _"""' __ ~ _____ , I I I I I I I
I I I I I I I I I I I I I I I I
( L _______ _
End
I I I I I I I I I
)
~--------- J
I
Flow chart of the stator optimization design
Fig. 9. 22
A desired optimization result is obtained as shown in Table 9. 2 and Table 9. 3. The difference between the two operating mode frequencies is only 12 Hz. Table 9.2
Structure parameters of the stator before and after optimization (Unit: mm) 11
12
b
bl
Other parameters
Before optimization
14.72
2
10
3.0
Same as Tahle 9. 1
After optimization
15.00
10
2.1
Same as Table 9. 1
Parameters
Values
Comparison of the operating frequencies before and after optimization (Unit: Hz)
Table 9.3
El mode frequencies
B2 mode frequencies
Differences
Before optimization
44 191
44042
149
After optimization
44 153
44 141
12
3. Experiment results The prototype motor is shown in Fig. 9. 23. The frequency response of is measured by using PSV-300F-B Doppler laser vibrometer, as Fig. 9. 21. The comparison between the experimental results and the results by the software ANSYS is shown in Table 9.1. It shows that
the stator shown in calculated there is a
Chapter 9
277
Linear Ultrasonic Motors
little difference between the calculated frequency and the experimental frequency. The reasons which lead to the difference arc: CDthc metal elastic body and the piezoelectric ceramic pieces arc taken as a whole in modal analysis, but actually the piezoelectric ceramic pieces are bonded to the metal elastic body; @ the discontinuity of materials and machining tolerance. Of course, the difference can be decreased further by using the method mentioned in Chaps. 5 and 7.
'1""[""1""[""1"" [""1""1""1""1'" 12
13
Fig. 9. 23 ~
= ~ ~
ts-
§
-0
" 8.
~[
20
14
15
16
Prototype motor
1 40
30 FrequencYIkHz
I
50
(a) Frequency response of second bendin g mode
40
30
50
FrequencYlkl-lz (b) Freq llency response of firsllongi tlldi nal mode
Fig. 9. 24
Measured frequency response of the stator
Table 9. 4 Calculated results and experimental results of the stator mode frequencies (Unit: Hz) Operating modes
Calculated results
Experimental results
Differences
-47
First longitudinal mode
44 153
44200
Second bending mode
11 111
13 930
211
12
270
-258
Difference
278
Ultrasonic Motors Technologies and Ap plicalions
According to the experiment results. the second bending mode of the stator can be measured directly by PSV-300F-B while the sine voltage with a frequency of 43. 93kHz is applied to the piezoelectric ceramic pieces on the side of the stator. Similarly. the first longitudinal mode of the stator can be observed while the sine voltage with a frequency of 11. 2kHz is applied to the piezoelectric ceramic piece at the surface of the stator. Under the pre-pressure of 30N and the excition voltage of 100V!'!'. the speed without load of the motor is 126mm/s. and the maximum thrust is 2:'\1.
9. 2. 2
Linear Ultrasonic Motor with Single Driving Foot
The motor described in Section 9. 2. 1 has some drawbacks. such as the bonded piezoelectric ceramic pieces to the surface leading to an increase of the stator volume. inconvenient to the clamping of the stator and the installation of the motor. However. it can be seen that it is possible to construct many kinds of linear ultrasonic motor using longitudinal and bending modes of the thin plate. In order to obtain the effective elliptical motion on the driving foot using least piezoelectric ceramIcs. a novel linear ultrasonic motor based on in-plane modes has been presented.
1. Operating mechanism (1) Structure of the motor Figure 9. 25 shows the structure of a new linear ultrasonic motor based on inplane modes using single driving foot. The stator is also a thin metal plate. The especial design of the stator is that the displacement in tangential direction is amplified in normal direction by a triangle structure of the stator while the longitudinal mode of the stator is excited. Only two rectangular piezoelectric ceramic pieces polarized in the thickness direction arc bonded to the side face of the stator. The stator is elamped in the housing by silicone rubber. A commercially available bar-slider is used as the mover and a friction lath is bonded to the surface of the bar-slider to enhance the wear resistance. The pre-pressure between the bar-slider and the stator is provided by bolt and silicone rubber. (2) Mode shapes of the stator This stator also utilizes the E1 and B2 modes as operating modes but with much
Fig. 9. 25
Structure of the motor
Chapter 9
279
Linear Ultrasonic Motors
difference from that discussed in Section 9. 2. 1. The modal analysis result of the stator is shown in Fig. 9. 26. Two vertexes of the triangle structure of the stator, points 1 and 2, are designed on the wave crest and trough of the B2 mode, respectively. In the case of the second bending mode is excited, as shown in Fig. 9. 26 (a), the point 1 moves down and the point 2 moves up alternately. which leads to a horizontal movement of the point 3. In the case of the first longitudinal mode is excited, as shown in Fig. 9. 26(b), the movement of the points 1 and 2 in horizontal directions is converted into the movement of the point 3 in perpendicular direction by the triangle structure. Thus. there will be an elliptical motion on the point 3 while the two operating modes are excited simultaneously.
(a) Second bending mode shape
Fig. 9. 26
(b) Firsl longiludinal mode shape
Operating modes of the stator
(3) Operating mechanism The excitation of the stator is shown in Fig. 9. 27. Two rectangular piezoelectric ceramic pieces polarized in the thickness direction (see Fig. 9. 27 (a» are bonded to the bottom of the stator (see Fig. 9. 27(b». With the excited mode as shown in Fig. 9. 27, the four essential states in the sequence of operating of the stator are shown in Fig. 9. 28. and described as follows:
- Whase B cosw/ (a)
( b)
Fig. 9. 27
Excitation of the stator
CD The displacement in rightward direction of the driving foot reaches a maximum when the stator reaches the largest deformation state of the bending mode. @ The displacement in upward direction of the driving foot reaches a maximum when the stator reaches the largest contraction state of the longitudinal mode. ® The displacement in leftward direction of the driving foot reaches a maximum. The stator returns to the balance state of the longitudinal mode and rea-
280
Ultrasonic Motors Technologies and Ap plicalions
ches the largest deformation state of the bending mode. @ The displacement in downward direction of the driving foot reaches a maximum when the stator reaches the largest streteh state of the longitudinal mode. Obviously, there is an elliptical motion on the driving foot. By repeating the sequences CD to @, the motion of the stator can be converted into the linear motion of the mover pressed on the driving foot. If the phase of the power source applied to the stator is inverted, the direetion of the elliptieal motion is reversed.
Fig. 9. 28
Operating mechanism
2. Experiment results Figure 9. 29 shows the prototypes of the stator and the motor. The mass of the stator is only 18g, and the dimension of the jacketed stator is 55mmX 26mmX 14mm. Experiment results are obtained as follows:
(3) Prolotype of 51310r
Fig. 9. 29
(b) Prololype of 111010r
Prototypes of the stator and the motor
(1) Frequency response of the stator The frequency response of the stator is shown in Fig. 9. 30. The second bending mode and the first longitudinal mode are excited when the driving frequeney is 51. 58kHz and 55kHz, respectively. The frequency difference is 120Hz. which can be farther redueed by the optimal design and experimental adjustments. Then the efficiency will be increased.
Chapter 9
Linear Ultrasonic Motors
281
~ 4.0
%3.0
j~54 . 58Hlz
/
~
"
.~ 2.0 C.
5kJ--lZ
\J
E < 1.0
52
50
Fig. 9. 30
54 56 Freq uencyfkHz
58
60
Frequency response of the stator
(2) Charactcristics of thc motor Undcr thc driving voltagc of 400VI'I" driving frcqucncy of 55kHz and diffcrcnt prc-prcssurc, thc mcchanical charactcristics and cfficicncy curvcs of thc motor are shown in Fig. 9. 31 and Fig. 9. 32, respectively. Obviously, the speed without load decreases and the thrust increases as the pre-pressure increases. When thc prc-prcssurc is 24N, thc maximum thrust is 2. 5N, thc thrust-wcight ratio is up to 11: 1, the maximum efficiency is up to 8 %. 200 ,------------------------------, 160
- )(-
Pre-pressllre ~ 1 6N
- A-
Pre-preSSllre~24
~ 120
12 .-----------------------------~
8
"E
I
4
5
40
L.l.J
Pre-pressure~ 1 6N
Pre-press llre~ 24N
\
./ /t:~
E
8.
'"
"-
~ .<:;
"-ti 80
X 11
'x
i
I
,-
,t¥
><;1
.x/ ' / l
IA
11
\~ ,
'?<
\
'&
\
11\
\
\
1~
°0~~------L-----~~ 2--------~3
Thrust!
Fig. 9. 31 Mechanical characteristics of the motor
9.3
Thrust!
Fig. 9. 32 EIIiciency curves of the motor
Linear Ultrasonic Motors Based on d33 Effect
As described previously, the linear ultrasonic motor based on d'l effect of piezoelectric ceramics has been developed rapidly due to its simple structure. Howevcr, thcir thrust is usually lcss than 10)J bccausc thc elcctro-mcchanical transformation efficiency using d'l effect is low. As we know, TE vibration type of the piezoelectric vibrator is the thickness extension-contraction vibration parallel to the electric field direction, which utilizes the d 33 effect of piezoelectric ceramics. The linear ultrasonic motor using d" effect has a higher electro-mechanical transformation efficiency than that of the aforementioned motors, although it possesscs largcr volumc.
282
Ultrasonic Motors Technologies and Ap plicalions
In this section. two bolt-clamped linear ultrasonic motors developed by PDLab arc used as an example to explain the motion mechanism of the linear ultrasonic motors using d" effect of PZT.
9.3. 1
Linear Ultrasonic Motor with Butterfly Shaped Stator- 27J
1. Operating mechanism (1) Structure of the motor The linear ultrasonic motor with a butterfly-shaped stator is developed by PDLab, as shown in Fig. 9. 33. It mainly consists of the stator and mover. The stator is a symmetry structure. Eight rectangular piezoelectric ceramic pieces polarized in the thickness direction are placed between the front-end block and rearend block and clamped by bolts. Two such piezoelectric ceramic pieces with opposite polarization direction arc set as one group and the bronze electrodes arc elamped between two piezoelectric ceramic pieces. The mounting block is used to mount the stator. The mover is a commercially available bar-slider. In order to enhance wear resistance. a friction lath is also bonded to the surface of the barslider.
Friction plate Bar-slider
Fig. 9. 33
Structure of linear ultrasonic motor with a butterfly shaped stator
(2) Mode shapes of the stator A symmetry mode and an anti-symmetry mode are utilized as operating modes. The modal analysis of the stator using A:'\JSYS software has been carried out and the result is shown in Fig. 9. 31. The symmetrical vibration mode (E,). as shown in Fig. 9. 34 (a), is formed while two piezoelectric vibrators (called left wing and right wing) arc excited symmetrically and simultaneously. The antisymmetry vibration mode (Eo)' as shown in Fig. 9. 31(b), is formed while two piezoelectric vibrators arc excited anti-symmetrically and simultaneously. The piezoelectric ceramic pieces are arranged on the both sides of the nodal plane of the longitudinal mode. In order to decrease the influence caused by clamping of the stator on the vibration mode of the stator, the node points 1 and 3 are taken as the clamping point of the stator. (3) Operating mechanism Figure 9. 35 shows the movement coordinates of the stator. The two piezoelec-
Chapter 9
Linear Ultrasonic Motors
(a) Symm etric mode (EJ
283
(b) Anti-symmetric lIlode{EJ
Fig. 9. 34
Operating modes of stator
tric vibrators stretch and shrink simultaneously when two pairs of piezoelectric ceramic components above the mounting block are driven by sinw t voltage signals. thus the symmetric vibration mode is excited. One piezoelectric vibrator stretches while the other one shrinks when two pairs of piezoelectric ceramic components underneath the mounting block are driven by cOSwt and - cOSwt voltage signals. respectively. thus the anti-symmetric vibration mode is excited. The two vibration modes would be excited simultaneously when the two voltage signals arc applied on the piezoelectric ceramic components at the same time. thus the elliptical motion of the particles on both driving feet is formed in the same direction.
yf
y
Fig. 9. 35
Sketch of stator movement coordinates
In order to simplify the analysis, the influence of the disturbing modes and the damp is not taken into account. The symmetric mode of the stator can be excited when one group of piezoelectric ceramic components both in the left and right wings is imposed by a sinwt voltage, as shown in Fig. 9. 31(a). In the coordinate systems xOy and x'O' y' as shown in Fig. 9. 35, the displacement responses of the coordina te origins 0 and 0' are shown in Figs. 9. 36 (a) and (d). From Fig. 9. 36 (a), the displacement response u, of the point 0 in the xDy coordinate system can be described as U, =
U,sinwt'e,
(9. 7)
where e, is the unit vector, a, is the included angle between e, and x axis. U, is the displacement response amplitude. is the ineludcd angle between two wings,
e
284
Ultrasonic Motors Technologies and Ap plicalions
land h are the length of the driving feet and the width of the beam, respectively. From Fig. 9. 36 (d). the displacement response u: of the point 0' in x'O' y' coordinate system can be described as
U:
u',
U: sinwt ee',
=
(9. 8)
where e', is the unit vector. is the response amplitude. The angle between the unit vector e: and .7:' axis is also a,. The anti-symmetric mode of the stator can be excited when voltages of cOSwt and - cOswt are applied to one group of piezoelectric ceramic pieces of both piezoelectric vibrators. respectively, as shown in Fig. 9. 34(b). The displacement responses of the coordinate originsOandO' arc shown in Figs. 9. 36(b) and(e). From Fig. 9. 36 (b). the displacement response u, of the point 0 in .,{)y coordinate system can be described as u,
U, coswtee,
=
(9. 9)
where e, is the unit vector, U, is the response amplitude. The angle between the uni t vector e, and .7: axis is a,. From Fig. 9. 36 (c), the displacement response u: of the point 0' in x' 0' y' coordinate system can be described as (9.10)
U:
e:
where is the unit vector, is the response amplitude. The angle between the unit vector eo' and x' axis in the negative direction is also aa' From Fig. 9. 36(e). there is the phase difference of rr/2 in time and the phase difference a, + a, in space between u, and U" If (a, + a, ) is rr/2, the motion track is a normal ellipse with a deflection angle a, between the short axis and the x aXIs. Also, from Fig. 9. 36(f), there is the phase difference of rr/2 in time and the phase difference of [rr- (a, + aJ ] in space between u; and u~, so the elose motion
"~ , '~ yf'
/,
~ o a,
x
y
(a) E, mode
(d) E. mode
x
x x' (c) Displacement resona nce o f the driving foot at 0 poin t
(I) Displacement resonance of the driving foot at O· poilll
y
y'
(b) E, mode
(e) E. mode
Fig. 9. 36
Displacement response of driving feet
Chapter 9
285
Linear Ultrasonic Motors
track is also formed on the driving foot. If [rc - (a, + a,) ] is rc/2. the motion track is also a normal ellipse with a deflection angle a, between the long axis and x' axis in negative direction. Therefore. if the two groups of drive voltages as shown in Fig. 9. 34 are applied to the corresponding groups of the piezoelectric ceramic pieces at the same time. respectively. the displacement response of the driving pointOinxOy coordinate system can be expressed as {
Ux :
U,sinwtc~sa, ~ U,coswt~osa,
Vy -
U, cOSwt Slna,
(9. 11)
U, Slnwt Slna,
Also. the displacement response of the driving point 0' in .L'O'y' coordinate system can be expressed as {
Ux'
=
v y'
= -
, U ',s.lnwt COSa, - U',coswt COSa,
(9.12)
U: sinwt sina, - U: cOswt sina,
The parameters of a, and a, can be adjusted to be 0 and rc/2 by adjusting the structure parameters: the angle {}. the length of driving feet l. the width of beam h. respectively. Thus. the motion tracks of both driving feet are normal ellipse. The displacement response of the driving points 0 and 0' can be expressed as {
u, si~wt U,coswt
Ux : Vy -
{
(9.13)
U: sinwt Vi = - U:coswt Ux'
=
(9.14)
The motion process of the stator in a cyele is shown in Fig. 9. 37. Eq. (9.13) and Eq. (9. 11) indicate that there is the phase difference of rc in the displacement response between the left and right driving feet. Therefore. two driving feet
Longitudinal vibration direction _ Movement direction of bar-slider
~
(1)\ .---------JIIW------Q?~-----. (4)
~
(2) \
\
~(3))
\
~
~ S
~
Fig. 9. 37
Operating mechanism of the motor
Ultrasonic Motors Technologies and Ap plicalions
286
push the slider alternately in a cycle.
2. Design of the stator (1) Sensitivity analysis of the stator mode frequency Figure 9. 38 shows the main structure parameters of the stator. The initial structure dimensions of the stator arc shown in Table 9. 5. The finite element model of the stator is set up based on the structure dimension of the stator shown in Table 9. 5. The sensitivity of the symmetric and anti-symmetric mode to each structure parameter is shown in Fig. 9. 39. Table 9.5
The stator initial dimension (Unit: mm)
Pz
Parameters
13
Values
15
19
10
8
Figure 9. 39 indicates that the relative sensitivity of the mode frequencies of E, and E, to the structure parameters. Obviously, the structure parameter P j can be selected to adjust and match the two mode frequencies.
;E
?;>
:~
'~ ~
··,···f··· .. ···.···,·· ·;····.···, ·· ·,· ···,· · ,
-9
'<;s - 12 0<:
•
•
•
,
I
•
,
I
I
..:- .. ; .. -:. .. ; ... , . .-:....:- .. , .. -: • E, .~ . .. ~ ...:... ~ ... : ... ~ ...:... : .. . : . E .
- 15 '--7:1', --I'::-,--1'::-3--J::-,,-~/:':" \-~J~ 'o...J
Sln,clure para meIer
Structure parameters of the stator
Fig. 9. 38
Sensitivity of operating mode frequencies to structure parameters
Fig. 9. 39
(2) Design for frequency consistency of the two operating modes According to the sensitivity analysis. the mode analysis is carried out by using the finite element analysis model of the stator with different structure parameter values of P j
•
The results are shown in Fig. 9.10. It can be found that the differ-
ence between the two operating modes is only 4Hz when the length of the rearend block is 13. 45mm. In order to obtain larger amplitude on the driving feet of the stator, the rearend block of the stator uses a phosphor bronze material and the front end block uses a duralumin material.
3. Experiment results Figure 9.11 shows a prototype of the linear ultrasonic motor with the butterfly shaped stator. Experiment results arc obtained as follows: (1) Frequency response of the stator
The measured frequency response in normal and tangential directions arc
Chapter 9
Linear Ultrasonic Motors
287
shown in Fig. 9.12 and Fig. 9. 13, respectively. The comparison between the experiment results and calculation results of the stator is shown in Table 9. 6. The ealculated values are larger than the experimental values.
501':1---~ 12:-----:'13:-----:1~ 4-----:-'15
""mm
Length of stator rear-end block versus mode frequency
Fig. 9. 41
Fig. 9. 40
:?
Prototype of the motor
:? 10
10
E
%
~
-c
.~ C.
1
5
<"
O ~==
C.
~ O40~~----------~------~ 45 50 55
____~____~________~
~
55 FrequencylkHz
FrequcncylkHz
Frequency response in normal direction of the stator
Frequency response in tang en tial direction of the stator
Fig. 9. 42
Fig. 9. 43
Table 9. 6
Comparison between calculated and experimental mode frequencies (Unit: Hz)
Operating modes
Calculated values
Experimental values
Differences
Symmetrical mode
52 124
49 310
2 814
Anti-symmetrical mode
52 120
19 270
2 850
4
40
-36
Frequency differences
(2) Characteristics of the motor Figure 9.44 shows the motor's speed-frequeney performance. Under the prepressure of 70='J, voltage with the amplitude of 200VI'I' and operating frequeney of 50kHz. the maximum speed of the motor is up to 780mm/s. Fig. 9.45 shows the meehanical charaeteristics under different pre-pressures. In the ease of the pre-pressure of 16='J. the maximum speed of the motor without load is 950mm/ s. The maximum thrust of the motor increases with the pre-pressure mcreasmg. The maximum thrust is 16N when the pre-pressure is 70='J.
Ultrasonic Motors Technologies and Ap plicalions
288
1000
1000
Pre·pressure=70N V..=200V
800
- ¢ - 16 --0- 40N
800
:w 600
.!('
E
E
~
'[ 400
""
"eo. en
-i:r- 70N
600 400
Q.
en
200 044
0
52 FrcqucncylkHz
Spccd vcrs us frequency
Fig. 9. 44
9. 3. 2
200 2
0
Fig. 9. 4S
6
8 10 12 Thrusl/N
14
16
18
Speed versus thrust
Linear Ultrasonic Motor with Wheel Shaped Stator"J
In 2005, Shuxiang Dong developed a small, linear ultrasonic eryo-motor using two orthogonal third bending modes of the rod L29J , which had only one wheel as driving foot. In the section above, it can be seen that the linear ultrasonic motor with butterfly-shaped stator can push the mover two times in a cycle. Thereby, an inspiration comes into our mind: to design a stator with two wheels which can drive the slider alternately in a cyele and thus the efficiency of the linear ultrasonic motor can be improved.
1. Operating mechanism (1) Structure of the stator
Figure 9.16 illustrates the wheel shaped stator. The stator is a notched cylinder with the diameter of 20mm and the length of 1l0mm. The eight piezoelectric ceramic rings used in the stator are 20mm in outer diameter, 8. 5mm in innerdiameter, and 1 mm in thickness. Piezoelectric ceramic rings are placed in both ends of the stator. Each piezoelectric ceramic ring has been coated with two segment electrodes and under which the ceramic segments arc poled in two reverse directions. The eight piezoelectric ceramic rings arc arranged as shown Fig. 9. 17. Two such piezoelectric ceramic rings are set as one group, and there are four groups. The two groups assembled outboard of the stator, group 1 and group 2, arc used to excite the fourth bending mode in y direction, and the other two groups assembled inside of the stator, group 3 and group 1, are used to exci te the fourth bending mode in :x; direction. Press;n gbloc k
EICClrodc
PZT I'
~ ~Right side
~ Reverse side
/
Screw cap
Phuse A ", __.~_---'"-<:-,,,...=-_ _ _--' SUpJXlft
Fig. 9. 46
Structure of stator and piezoelectric ceramic ring
Chapter 9
289
Linear Ultrasonic Motors
r - - - - - - - - - - - - . - -.... Phase B Phase A- - - - , - - - - - + - - - - - - - - - - + - - - - - - - ,
Electrode
Group I
Group 4
I t ~~---~---~--~---L----~--~
l"ig. 'I. 47
Arrangement ot the plezoelectnc ceramIc nngs
(2) Modes of the stator As we know. the non-uniform beam is of ten used to amplify the amp Ii tude in designing a transducer. Similarly, as shown in Fig. 9.46, the role of the notches in the stator is to decrease the effective bending stiffness of the stator. Fig. 9. 48 shows the modal analysis of the notched stator using FEM. Clearly, excitation of the two bending modes of the stator results in a much larger bending amplitude at the stator's driving feet.
,f-z x
(b) Bending mode in y direction
(a) Bending mode in x direction
Location of right driving foot
0,5
Location ofPZTs
"'"
~
y direction
______ x direction
o 109
- 05
-I
Location of left driving foot (c) Displacement mode shape
Fig. 9. 48
Mode analysis of the stator using A:"ISYS software
(3) Operating mechanism The motion of the stator in a cycle is shown in Fig. 9. 49. Just like the linear ultrasonic motor with butterfly-shaped stator, the elliptical motion can be obtained on the stator's driving feet and the slider pressed on the stator will be driven by the two driving feet alternatcly. Thus, the stator drives the slider two times in one circle.
Ultrasonic Motors Technologies and Ap plicalions
290
(" d ' (d)
(c)
Operating principle of the linear ultrasonic motor with a wheel-shaped stator
Fig. 9. 49
2. Design of the motor Three key problems combined with the design processes of the motor are: clamping of the stator with little influence on its operating modes; mounting of the stator to ensure both the high stiffness in the direction of motion and elasticity in the direction along which the slider is pressed against the stator; simple structure of the motor. Based on an overall consideration of the above problems, a support plate with flexible hinge is designed to clamp the stator, as shown in Fig. 9. 50. Support plate with nexible hinge
Clamping plate
(a)
Fig. 9. 50
(b)
Support plate with flexible hinge (a) and clamping way of the stator (b)
The support plate with flexible hinge consists of three parts: the clamping plate which has large stiffness, leaf spring and grip block. The stator is fixed to the clamping plate by bolts. The support plate with flexible hinge is fixed to the basement, so that the driving feet uniformly contact the friction lath as shown in Fig. 9. S1. The pre-pressure between the stator and the slider is applied by belleville spring.
3. Experiment results The prototype stator and motor arc shown in Fig. 9. S2. The mass of the stator is about 22Sg, and the whole dimension of the jacketed stator is 120 mmX so mmX 62 mm, as shown in Fig. 9. 52 (b). Experiment results are presented as follows. (1) Frequency response of the stator
Chapter 9
(a) Prototype of stator
Fig. 9. 52
291
Linear Ultrasonic Motors
(b) Prototype of motor
Prototypes of stator and motor
The frequency response of the stator is shown in Fig. 9. 53. The fourth bending mode in y direction and in x direction are excited when the driving frequency is 23. 08kHz and 22. 78kHz, respectively. In theory, the frequency difference of the two isomorphic modes is zero. However. there is frequency difference 300 Hz because of the support which causes the different boundary conditions in two directions of the stator and heterogeneous materials. In order to excite the two orthogonal modes, an intermediate frequency of 22. 93kHz is utilized as driving frequency.
~
2.0
i"32.3 l kHz
i"23.08kHz
~ 1. 5 ] 1.0
f"22.78kHz
fo;32.25kHz
g...;: 0.5 ':'"L-::::::::....L..::::;::==~::=::::...L.:::::::::;=__.....I 30
35
FrequcncylkHz
Frequcl1 cylk l--l z
(a) Frequency resonance in y direction
Fig. 9. 53
(b) Freq uency reso nance in .r direction
Frequency resonance of the stator
(2) Characteristics of the motor The speed-frequency characteristic of the motor is shown in Fig. 9. 54. Under the driving voltage of 200VI'I" and the pre-pressure of 70N. the maximal speed of the motor is 346mm/ s while the driving frequency is 22. 8kHz. The load char-
292
Ultrasonic Motors Technologies and Ap plicalions
acteristics of the motor are shown in Fig. 9. SS. The maximum thrust is 19N. the thrust-weight ratio is 8. 4, and the maximum efficieney is 23 %. Obviously. the no load speed decreases with the load increasing. 400
~E ~
:lCo
V>
•
300
400 , - - - - - - - - - - , - ----,. 0,3
! i
0,25
II
0.2
"'-"'-"-l-"'--"-'"
200
0,15 ~
100 0 22 .5
;J<
0 0
22 .8 23. 1 FrequellcylkHz
23.4
10
"
0,1
ii .<:;
0.05
LlJ
E
15
ThrustIN
Fig. 9. 54 Measuring speed-frequency characteristic of the motor
Fig. 9. 55
Measuring load
characteristics of the motor
Contact Model of Standing Wave Type LUSM
9.4
In 1993, Ueha and Tomikawa proposed the contact model for LTUML30- • which has been introduced in detail in Section 8. 2. 3. The linear ultrasonic motor described in this chapter are also hybrid type transducer. which have much similarity with the L TUM. Based on the research of force transmission model of L TUM, a contact model is developed for understanding the operating mechanism of the linear ultrasonic motor.
9. 4. 1
Steady State Characteristics
Based on the analysis in Section 8. 3, the thrust in steady state can be described. When 0 < ({J< 21(, there are two conditions: one is that the equal velocity point is in the range of the contact area; the other is that the equal velocity point exceeds the range of the contact area. Taking them together. the thrust can be written as
(2
.
. .!£.] (9.1Sa) • . w)' - v;n + arcsIn -Vrn- +.!£. - 1( ) cos .!£. - sm [ 2J(U, Ux w Ux w 2 2 2 if 0 < ({J < 2arccos( v,jU, w) or 1( ~ ({J < 21( F out
=
kl k~Uc [U
4
W
+ ;U~ [sin f (1d
(({J -
sin({J) -
Vrn (
sin f
(9.1Sb)
- fcos f J
if 2arccos( vrn/Ux w) ~
({J
<
- fcos f ) ]
1(
where U x and U y are the vibration amplitudes of the point on driving foot in tangential and normal directions, respectively;
Vrn
is the speed of the slider in steady
Chapter 9
293
Linear Ultrasonic Motors
state; other symbols refer to Section 8. 3. When rp = 21(,
where Po is the pre-pressure.
9. 4. 2
Transient Responses
As we know, an ultrasonic motor has the characteristic of quick response. The start-up time, in which the slider changes from static state to steady state, IS m milliseconds. Obviously, the start-up time is much longer than an vibration eyele of the stator because the operating frequency of the linear ultrasonic motor is above 20kHz. The motion equation of the slider can be described as (9.17) where m, v, F d , and Flood are the mass of the slider, the speed of the slider, the driving force of the stator, and the load, respectively. In the start-up process of the slider, the conditions of the motor can be described as {
t =
t ;?c
0, v
'0 , v
= =
0, Fd v""
F m"
=
Fd
=
(9.18)
Flood
where V m ' F m " , and '0 indicate the no-load speed in steady state, the maximum driving force, and the start-up time, respectively. Hence, the driving force can be written as (9.19) Substituting Eq. (9.19) into Eq. (9.17), the speed of slider can be written as v
=
where the start-up time factor
v", [ 1 -
'0
exp (- ,to ) ]
(9.20)
is given by
'0
=
mVrn
(9.21)
=-------"':c:--
F 10ad
Fmax -
Then, after the power is cut off, the load F, brake the motion of the slider. mv(t) = -
Flood
and the interface friction force
F, - F locd
(9.22)
where the interface friction force F, is given by (9.23) In the shutdown process of the motor, the condition can be described as {
t = t =
0 ,v
=
'1 ,v =
Vm
0
(9.24)
Ultrasonic Motors Technologies and Ap plicalions
294
The speed of the slider after power cut off and the shutdown time factor be written as follows. respectively. v
=
'1 - '1
v", exp ( - t / )
/l-d
Po + m
Flood
[1
-
'1
exp ( - / t )]
'j can
(9.25) (9.26)
9. 4. 3
Simulation Examples
Here. the linear ultrasonic motor with a butterfly-shaped stator and linear ultrasonic motor based on in-plane modes described in Section 9. 2. 2 and in Section 9. 3. 1 have been used as examples to validate the contact model. Parameters needed in the simulation are shown in Table 9. 7. Table 9. 7
Simulation parameters of linear ultrasonic motors
Parameters Operating mode frequency/Hz
Butterfly shaped linear USM 49.5XIO'
Linear USM using in-plane modes 55.0XIO'
Normal amplitude/ f1m
1. 2
O.
3~0.
5
Tangential amplitude/f1m
2. 5
0.1~0.
5
Static friction coefficient
O. 15
0.1 O. 05
Viscous friction coefficient
O. 05
Young's modulus/Pa
7. 6e-10
2.06e-ll
Width of the friction laycr/mm
3.0
3.0
Contact area/m'
1.0c-5
5.0e-6
1. Simulation of mechanical characteristics Based on the analysis above. the mechanical characteristics of the linear ultrasonic motor arc predicted. Figures 9. 56(a)-(c) illustrate the influence of the friction material on the mechanical characteristics. Fig. 9. 56(a) shows the influence of the viscous friction coefficient on the mechanical characteristics. Obviously. the no-load speed docs not be affected by the viscous friction coefficient but the maximal thrust increases with it increasing. Fig. 9. 56 (b) shows the mechanical characteristics under different contact stiffness. It indicates that the mechanical characteristics become "hard" with the increase of the contact stiffness. Fig. 9. 56 (c) shows the influence of the static friction coefficient on the mechanical characteristics. Obviously. the maximal thrust increases but the no-load speed docs not change with the increase of the static friction coefficient. Fig. 9. 56(d) illustrates the influence of the pre-pressure on the mechanical characteristics. Similarly. the pre-pressure has much influence on the maximal thrust but little influence on the no-load speed. In order to validate the contact model, the simulation results and the corresponding experimental results arc compared in Fig. 9. 57. 2. Simulation of transient responses From Eqs. (9.20) and (9.25). the transient responses of the butterfly-shaped
Chapter 9
Linear Ultrasonic Motors
295
0.8 0.6
~ ~
0.4
1!0-
CIl
°0~--~-~1~0-~~1~5--~20
Thru stIN
ThnlstIN
(a) Mechanical characteristics under different viscous fric tion coefficient 0.8 ,--------,,---------.-------.-----,
..
0.6
~
i
:--,_ ..........
0.4
I
:
!I,~O
•
~i
\ i
' ;
1! 0-
CIl
0.2
.
\
r- -
~ OA
.V. . . ~ . ...j.
; 0
°0~--~----~IO--~~I)~ ·~~ 20
15 10 Thrust (d) MechaniC<11characterist ics under different pre·pressure
0
ThrustIN (c) Mechan ical characteristics under different static friction coefficient
Fig. 9. 56
0,2 ,..---.----,---~--.-----,,---..,
0.7 ':~~"'o-~.:.:~:~i
~
0.5
0-
0.3
t CIl
0.4
-1
Experiment resul1
0
: : : : : : : : !.~.~:=~:::~ C:~:: : : I,: : : : : : : :
··............·i .. ·····......··(~\.. \ ·........ ·( ....·· ..·.. · •
=
-----~
it
=
·r····· ..'·..
10 15 20 TImlstIN (a) Butterfly·shaped linear ultrasonic motor
Fig. 9. 57
----. Calculated resuhs Po=24N
Calculated results
0.2 ················~················+······i········+···· .......... . : ~ \ ~ 0.1 ················!·············· .. o··r······· ....... . 00
20
Simulation results of butterfly·shaped linear USM
0.8
0.6
Pr 70N
JP.~OON
\
%
•
. '
....
0.6
,
l"
0,2 . .•• 10 . , . . !'.~0 . 15 - - ~.~0 , 20
---- P.- 15N -·-'- I P.~30N
: " ! "\ ; ~ ~ ~ ! \ ; i, ...'~· ··· ···1: ....... ... --y-......•... •
CIl
0.8
'
············~······ 1······~··.·· · · · ··· :
0-
(b) Mechanical characteristics under different contact stiffness
=-_ .........
0.1
t
~
.~-."O--7-"Q._....
0.1
-
o
experiment results
Calculated reslJhs Po· 16 experiment results
-4
.
~ o·~ ··;·+···i
+.. . . . .+.........
11
···········f·········::r···" ...:J;. ......
1.5 2 2.5 TI.ru tIN (b) Linear ultrasonic motor using in-plane modes 00
0.5
Comparison between the simulation and experiment results
linear ultrasonic motor are predicted, as shown in Fig. 9. 58. It can be seen that the start-up time is much larger than the shutdown time and with the load increasing. the no-load speed in steady state decreases but the start-up time increases. Furthermore, with certain load, the start-up time increases with the increase of the mass of the slider.
296
Ultrasonic Motors Technologies and Ap plicalions
0.5
0.4 ~
~1l
"'-
OJ 0.2
en
0.2
0. 1
20
40
60
/;:S.:
t/··;-~;'-:iI-
if J
_
l i
·f··········_·
!
Time/ms
9.5
m=5e-2kg m=5e-3kg
°0~--~2~0----~ 40~--~ 60~--~8~O~
80
Tim
(a) Transient responses for different load
Fig. 9. 58
:
111S
(b) Transient responses for different mass of slider
Transient responses of butterfly-shaped linear ultrasonic motor
Synergetic Operating Technique of LUSMs
In some application, the output force of a linear ultrasonic motor can not meet need. Therefore, an idea of synergetic operating technique comes into being. The output force of the motor will increase after the synergetic operating technique is adopted, but the efficiency will decrease. The main reason is that the stator has different dynamic characteristics even though they were fabricated with the same design. Moreover, there will be a mutual coupling phenomenon when several stators are operated at the same frequency and result in the dissipation of energy. Mraeek et al. had done some preliminary research on how to synergetic operating of linear ultrasonic motors L32 -. As shown in Fig. 9. 59(a), a bundle of motors consist of four stators drive one slider simultaneously. Each stator consists of two piezo vibrators. The frequency response of eight same type piezo vibrators are depicted in Fig. 9. 59 (b) , I Y I of which indicates the admittance magnitude. The resonance frequencies of the eight piezo vibrators differ a little. Based on
,
<>
>V::::::::::~~
~>-. FrequencylkHz (a) Synergetic operarion of line.,r ultrasonic motors
Fig. 9. 59
(b) Frequency response obtained by ingle piezoelectric vibrator
Synergetic operating of linear ultrasonic motors and frequency response
Chapter 9
297
Linear Ultrasonic Motors
the considerations for driving a single motor, Mracek, Hemsel, Wallaschek, et at. evolved four main driving strategies for a set of motors as shown in Fig. 9. 60:
8L3
8 1. 4
81.5
81.6
81.7
8 1. 8
81.9
8L3
8 1.4
Fig. 9. 60
8 1. 5
81.6
81.7
Frequency /kHz
(a) Individual exci lalion
(b) Single resonalll excitation
"I o
81.3
81.4
FrequencylkHz
81.5
81.6
81.7
81.8
81.9
·········i
81.8
81.9
........ j.......... j........ _-;.......... j..... .
80 80.2 80.4 80.6 80.8 81 81.2 81.4 81.6 81.8
Freq ucncyik Hz
Frequency /kHz
(c) Sweep excitation
(d) Single non·reSOnanl .,citalion
Driving method of synergetic operating of linear ultrasonic motors
(1) Individual excitation: as shown in Fig. 9. 60(a), every vibrator is excited by its own resonance frequency. A complex "beat" coupling phenomenon will be produced when the four stators work together on the slider with closed-spaced vibration frequency. The stator will also produce "beat" phenomenon because of the stator is excited by other stator through the slider in the adjacent driving frequency. This behavior will result in the instability of the running of the slider and lower efficiency of the motor. (2) Single resonant excitation: all vibrators will be driven at a single excitation frequency within the resonance area of all vibrators, as shown in Fig. 9. 60 (b). This approach can avoid the coupling phenomenon. However, it will aeeclerate the wear and also lower the efficiency of the motor for the driving velocity of each stator is different. (3) Sweep excitation: as is depicted in Fig. 9. 60(e), a bandwidth in the resonance area of all vibrators will be defined and the excitation signal will be swept up and down in frequency. Operating in this style, each stator has the same vibration period. there will be no "beat" phenomenon, and the contribution of each stator for driving the slider is similar. However, the energy diffusion problem has not been solved yet because the tangential velocity of each stator is different at random time. (1) Single non-resonant excitation: Equal to the "Single resonance excitation"
but in this case the excitation frequency will be chosen in the non-resonant area
298
Ultrasonic Motors Technologies and Ap plicalions
of all frequency responses. as shown in Fig. 9. 60 (d). In contrast to individual excitation strategy. an excitation at a single frequency in the non-resonant area would be very simple and stable, but of low efficiency. Furthermore, the amplitude of each stator will decrease largely if the driving frequency departures from the resonant frequency too much. In conclusion. the main problem for synergetic operating technique of linear ultrasonic motors is to improve machining quality and ensure close dynamic characteristics of every vibrator. Now, the synergetic operating technique of linear ultrasonic motors has been used step by step. Fig. 9. 61 shows an application in a large-scale astronomical telescope system. in which three transducers developed by Kurosawa as shown in Fig. 9. 6 have been used.
Speed (no· load): 0.8 8m/s; Torque(max.): 8.3N· 1ll
Fig. 9. 61
Application of thc synergetic operating of linear ultrasonic motors
References [ 1J
[ 2J
Jian Liu. Study on Linear Ultrasonic Motor Based on In-plane Vibration Modes. Dissertation for the Degree of Master. "fanjing: Nanjing University of Aeronautics and Astronautics, 2001. (in Chinese) Chaodong Li. Research on Longitudinal and Bending Vibration Linear Ultrasonic Motor with Large Thrust. Dissertation for the Degree of Doctor of Philosophy. "fanjing: Nanjing
University of Aeronautics and Astronautics, 1999. (in Chinese) [ 3J
T Sashida, T Press, 2002.
[ 4J
T Takano, Y Tomikawa. Characteristics of the ultrasonic linear motor using radial and non-axisymmetric vibration modes of an annular plate. Jpn. J. Appl. Phys, 1995, 31Partl(9B): 5288-5291.
[ 5J
Kenjo. An Introduction to Ultrasonic Motors.
USA: Oxford University
W Wishnewskiy, S Kovalev, 0 Vyshnevskyy. New Ultrasonic Piezoelectric Actuator for Nanopositioning. Bremen, 2001.
[ 6J [ 7J
T Wakai, M K Kurosawa, T Higuchi. Transducer for an ultrasonic linear motor with flexible driving part. IEEE Ultrasonic Symposium, 1998(1) :683-686. Ultra High Vacuum Systcm for Chcmical Analysis and Thickncss Measurcment. [2007-0615]. hup: / /www.baysidemotion.com/web/BMGHome. nsf.
[ 8J [ 9J [10J
[2006-08-06]. hup: / /www. rockwell. eom/anorad/produets/airbearing systems/ntype/n250. html. Squiggle motor ovcrvicw. [2007-6-6]. http://www.ncwscaletcch.com/downloads.htm!. R Yoshita, Y Okamoto. Micro-piezoelectric actuator. Journal of Precision Engineering Society, 2002, 68(5): 615-618. (in Japanese)
Chapter 9 [llJ
Linear Ultrasonic Motors
299
M Kuwana, T Kanbara, M Tikami. Driving device using piezoelectric actuator. Proceedings of Spring Symposium of Precision Engineering Society, Japan, 1999: 311. (in Japanese)
[l2J
Weidong Liu, Sbiebun Di, Wansbeng Zbao, et al. Design and analysis of linear bipolar ultrasonic motor. Piezoelectrics & Acuustooptics, 1997, 19(4): 226-230. (in Cbinese)
[13J
Chenglin Gu, Gan Dong. Double IT type linear piezoelectric ultrasonic motor. Proceedings of the CSEE, 1998, 18(2): 226-330. (in Chinese) Chaodong Li, Hua Yao, Renqing Pei, et al. Small-sized bionic [Dot ultrasonic linear motor. Small & Special Machines, 2001, 11(6): 10-11. (in Chinese)
[14J [lSJ [16J
Wei Hu. Study on Linear traveling Wave Ultrasonic Motors. Dissertation for the Degree of Master. :'-Ianjing: :'-Ianjing University of Aeronautics and Astronautics, 1996. (in Chinese) Wei Hu, Chunsheng Zhao. Study on linear traveling wave ultrasonic motors. Journal of & Diagnosis, 1996, 16(3): 8-14. (in Chinese)
Vibration, Measurement
[l7J
Chaodong Li, Long Jin, Chunsheng Zhao. The characteristics of hybrid transducer type linear ultrasonic motor with large thrust and large stroke. Acta Acustica, 1999, 24 (6): 653-
[l8J
Chunsheng Zhao, Jian Liu. Linear ultrasonic motor and its vibrator. Patent, ZL9811128. 2, 1998-05-07. (in Chinese)
[19J
Jian Liu, Chunsheng Zhao. Study on the linear ultrasonic motor based on the vibration plane of the rectangular plate. Acta Acustica, 2003, 28 (1): 86-90. (in Chinese)
[20J
Jian Liu, Chunsheng Zhao. Design of the linear ultrasonic motor based on the vibration In plane o[ the rectangular plate. New Progress on Vibration and Wave Technology. Shenyang: Northeastern University Press, 2000: 255-259. (in Chinese) Chunsheng Zhao, Jian Liu. Linear ultrasonic motor based on the vibration in plane of the rectangular plate. Chinese Invention Patent, ZL01l27038. 1, 2001-07-27. (in Chinese)
651. (in Chinese)
[21J [22J
Chinese Invention In
Guoqing Huang. Research on a Longitudinal-bending Vibration Coupled Type Linear Ultrasunic Motor with Two Statur and Precision Stage. Dissertation [or the Degree of Master.
[23J
Nanjing: Nanjing University of Aeronautics and Astronautics, 2001. (in Chinese) Qunting Liu. Research on Longitudinal-bending Vibration Coupled Type Linear Ultrasonic Motor with Multi-stators. Dissertation [or the Degree o[ Master. Nanjing: Nanjing University
[21J
of Aeronautics and Astronautics, 2001. (in Chinese) Weiqing Huang, Qunting Liu, Chunsheng Zhao. Research on a working stage driven by linear ultrasonic motor. Small & Special Machines, 2004(3): 17-18. (in Chinese)
[25J [26J
[27J
Chunsheng Zhao, Jiamei Jin. Square plate type linear ultrasonic motor and its excited mode. Chinese Invention Patent, CN20071002096S. 7, 2007-01-0S. (in Chinese) Dong Yang. Research on V Shaped Linear Ultrasonic Motor with Double Amplitude Transformer. Dissertation [or the Degree o[ Master. Nanjing: Nanjing University o[ Aeronautics &. Astronautics, 2009. (in Chinese) Chunsheng Zhao, Yubao Li. A butterfly type linear ultrasonic motor and its excited mode. Chinese Invention Patent, CN200710021372.8. 2007-06-10. (in Chinese)
[28J
[29J [30J
[31] [32J
Yunlai Shi, Hanlei Zhang, Chunsheng Zhao, et al. Two DOF positioning stage using linear ultrasonic motors. Transactiuns of .1'Vanjing [lniversity oj Aeronautics & Astrunautics, 2008,25(3): 161-168. Shuxiang Dong, Li Yan, :'-Iaigang Wang, et al. A small, linear, piezoelectric ultrasonic eryomotor. Applied Physics Letters, 2005,86, 05350l. S Ueha, Y Tomikawa. Ultrasonic Motors Theory and Applications. Ox[ord: Ox[ord Science Publications, 1993. J C Piedboeuf, J D Carufel, R Hurteau. Friction and stick-slip in robots: simulation and experimentation. Multibody System Dynamics, 2000 (4): 341-354. M Mracek, T Hemsel, J Wallasehek. Synergetic operation of ultrasonic linear motors. The First International Workshop on Ultrasonic Motors and Actuators, 2005 (11): 23-21.
Chapter 10
Step Ultrasonic Motors Great research progress for step USMs has been achieved since their invention more than 10 years ago. In 1991, Kusakabe developed a standing wave and selfcorrection USM[1:, which made the USM succeed in step motion without feedback. Later, Miyazawa put forward a step USM[2 ': using a shifting standing wave mode based on lijima's standing wave USM in 1993. Furthermore, in 1999, the author and Guiqing Wang, et al. designed and fabricated a new selfcorrection type USM[1-5:. In the same year, the author and Long J in, et al. developed a mode rotary type step USM: 67J with 80, 120, and 168 steps in one eirele based on the theory proposed by Miyazawa. In 2000, Snitka designed an ultrasonic aetuator: 8: based on a linear USM using two modes, whose positioning accuracy reached nano-meter level. In 2003, Yong Jin, Jifeng Guo, et al. made a step USM L9 J by combining a longitudinal mode and a torsional mode. In the next year, Xiangeheng Chu made a shaking-head type step USM: lOJ . In 2005, the author and Jiamei Jin, et al. developed a self-correction type step USM using modal rotation- 11J , a mode alternation type step USM_ 12 -13 J , and a linear type step USM and a rotary type step USM using vibrator alternation- 11 - 15J . Several prototypes were fabricated in different sizes for each type. Compared to an electromagnetic step motor, the step USM has good characteristies, such as simpler structure, smaller size, better environmental adaptability, and no-electromagnetic interferenee1l6 -. It can be widely applied to optical devices, robotics, space shuttles, automatic control systems, military facilities, medical equipment and so on. The nano-meter step USM can also be used in electron beams, ion beams, X-ray, scanning electron microscope positioning, etc. A step USM can be described as an ultrasonic motor which can realize the step motion with a certain step size. USMs inelude two major types: a traveling wave and a standing wave, and these two types of USMs operate in two different ways. The traveling wave USM has good controllability, stability, and long life- 17 -22J , while the standing wave USM has different performance and can realize movement with different forms[20 3<J. Thus, many standing wave USMs arc made by different operating modes and different driving schemes. The step USM is a significant branch of USMs. The step USMs can be divided into the adjustable steplength type and the fixed steplength type. The adjustable steplength USM is driven step by step by switching onloff the driving signal in a carefully designed sequence. Its step length is decided by the
C. Zhao, Ultrasonic Motors © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011
Chapter 10
Step Ultrasonic Motors
301
width of the driving pulse L35 -. When the USM starts or shuts down, fluctuations of the velocity will affect the uniformity of every step. When every step displacement is smaller, this fluctuation will be not negligible. In the condition of longstep and thc multi-stcp positioning, the fccdback loops havc to bc applied in order to reduce the position error. The fixed step length USMs possess some special structural forms to obtain step motion in the open-loop control c'6:. This kind of step USMs has no accumulated error in multi-step operating. The single-step operating error is produced mainly from the machine work and assembly error.
10. 1
Step Control of USM
An adjustable steplength USM is controlled by switching onloff the driving signal. Theoretically, the electromagnetic motors can also realize the step motion by similar operating, but the control is more difficult because the coil motor can not very quickly response to the control signal and can not make self-locking when its power is shut down. USMs possess characteristics of self-locking, rapid response and high position resolution. These naturally present advantages for the control of accurate step positioning. USM's self-locking feature comes from the operating principle of friction drive. Usually, in the operating state, the contact area between the stator and rotor (or silder) is smaller than that in the non-operating state. Besides when it operates, there is also local sliding on the frictional interface, which makes USM's self-locking torque (or force) greater than the stalling torque (or the maximum output force). In other words, with the power shutdown, USM's rotor (or slider) will be quickly locked at the position where it arrives. This advantage is beneficial to the control of precision positioning. The rapid response means the stator will soon achieve the steady state after power on. Namely, USM can reach its rating speed quickly. USM's high positioning resolution is due to its micro vibration amplitude and high operating frequency. Generally, displacement in the sub-micron or nanometer level can be achieved when the stator makes the rotor (or slider) move in a vibration eyele. Plenty of experiments show that large fluctuation of the positioning accuracy appears at the period of starting and ending the motor.
10.1.1
Startup and Shutdown Characteristics of USM
Under normal circumstances, in order to increase the amplitude and reduce the energy consumption, the driving frequency is designed to be elose to the resonance frequency of the stator. In this way, the dynamic characteristics of USM is a key factor in precision positioning. The process, from the beginning of the stator's vibration to the accumulation of the energy and finally to its stable operating state, is called as the starting state of the motor. On the other hand, the
302
Ultrasonic Motors Technologies and Ap plicalions
process, from the moment when USM's power is off and the stator obtains its own initial condition of vibration to the momcnt whcn thc vibration of the stator stops completely and finally to states of shutdown the rotor (or slider), is called as statc of shutdown of the motor. The intcrmcdiate statc betwecn thcse two states is the steady opcrating statc of USM.
1. Stator's vibration response According to the vibration thcory dcscribcd in Chap. 4, the forccd vibration of thc stator can bc divided into thrcc stages: CDa startup(beginning of an cxcitation); CZ)a steady state(kccping thc excitation); @an attenuation(switching thc excitation). Here, stage CD and ® will be discussed because the vibration characteristics of thesc two stages directly rclate to USM's response speed and positioning resolution. Thc Ref. [37J has analyzed thc problem in detail and obtained thc conclusion that the time for USM's stator vibrating in the stage CD and ® is different. The diffcrcncc bctween thc two stagcs' time is mainly dctermincd by damping of the stator. The greater thc damping is, thc biggcr thc diffcrence will bc. Gcnerally, the damping of USM's stator is relatively small. So, the difference between the two stagcs' time is not large. It is not enough to only consider the viscosity damping of the stator. In actual circumstancc other damping factors still cxist. From thc point view of energy, it is valid that in the beginning of the forced vibration the damping prevents the system from accumulating energy. The greater the damping is, the longer the time t, to reach the steady-state will be. In the decaying stage, the damping will consume cncrgy. In other words, the grcater the damping is, the sooner thc decaying stage will bc, thc shortcr the timc t, to stop thc vibration will be. Consequently when the damping is bigger, the difference between the two stages' time 6.t = t, - t, will be greater. 2. The actual operating process of the motor Corresponding to the vibration of the stator, the actual operating process of USM can be also divided into three stages: CD startup stage; CZ) steady operating stage; @shutdown stagc. The characteristics of startup stage and shutdown stage for a typical traveling wavc USM arc shown in Fig. 14. 12 and Fig. 14. 13. In the startup stage of USM, there is some delay, from the beginning of the vibration of the stator to the starting of the output of the rotor's shaft, which is callcd mechanical hystercsis of the motor. This is due to thc wholc system's damping and stiffness of the shaft coupling. Whcn the rotor is asscmblcd, thc boundary conditions of the stator will change. It's equivalent to adding the constraint to the stator, which increases its natural frcquency and dccreascs its amplitudc. When the stator's amplitude dcclincs, thc pre-pressurc will increasc the contact arca on the contact interfacc between the stator and rotor at the same time. :'\Jot only the transmittability of thc tangential forcc on the contact interface is incrcascd, but also thc arca of sliding friction is increased, resulting in more energy dissipation. Thereby, the startup time increascs with thc increase of prc-prcssurc, whilc thc shutdown timc
Chapter 10
Step Ultrasonic Motors
303
decreases with the increase of pre-pressure. Thc startup and shutdown characteristics for TRUM-4S under different prcpressurc arc shown in Fig. 10. 1. It illustrates that thc startup time incrcases with the pre-pressure going up while the shutdown time declines slightly with the pre-pressure nsmg up, as shown in Fig. 10. 1. 1)
r----------------------------------,
.
-
• Startup
•
II~
"0Il
__
~
___ L_ _
~
220
_ _ _ _L __ _
~
Pre-~lr~~sLJrl'
_ __ L_ _
"btl
2~1I
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~
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~
280
'-N
Fig. 10. 1 Startup and shutdown characteristics of TRUM-15 under different pre-pressure
10. 1. 2
Step Control for USM
Based on the preceding analysis, we can conclude that besides the load, other factors influencing the startup and shutdown time come from the motor itself. Among them thc most unstable factor is thc frictional intcrfacc betwecn thc stator and rotor. When the USM runs, the frictional properties at the interface will change from time to timc duc to wcar, hcating, and other rcasons, which lcads to fluctuation of the startup and shutdown time. These are the main reasons that makc USM instability whcn it starts up or shuts down. Figure 10. 2 shows test results regarding to the angular displacement versus thc powcron time for TRUM-45. Fig. 10. 2 (a) shows that the poweron time is 10ms, and the fluctuation of the displacement is large. It is primarily due to the startup timc of thc motor is about 8ms. Besides, therc is the ovcrswing, which needs about 30ms to make the speed relatively stable. The situation will be better when the poweron time more than 10ms, as shown in the Fig. 10. 2(b). The fluctuation reduces significantly and good linear relationship can be observed when thc powcron timc more than lOOms, as shown in Fig. 10. 2(c). Another test is regarding to the motor's repeatability. As shown in the Fig. 10. 3, the USM was rcpcatedly tcstcd 10 times. Thc maximal dcgrcc of decentralization based on the arithmetical mean value demonstrates that the shorter thc powcron timc is, the worse thc repeatability is. When it turns in thc counterclockwise direction, the degree of decentralization is lower than 10% if the timc is morc than ISms. Whcn it turns in the clockwise direction, the dcgrcc of decentralization is lower than 10 % if the time is more than 25ms. For this USM, the diffcrcncc of deccntralization in the two directions is mainly causcd by the speed difference in these two directions of the USM. Further, the degree of thc decentralization shows the position error of thc timing displacemcnt. Thc
Ultrasonic Motors Technologies and Ap plicalions
304
longer the electrical power is applied. the longer the step of the USM is, and the smaller the position error is. 14,---------------------------, • Coulllerclockwise • Clockwise
>::'
"1l" E
'" ~ :;:; ~,;;, ~
~
"'
4
~ ~
2 01
2
4
3
~
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...."
>::'
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5
7
6
8
9
10
90 80 70 60 50 40 30 20 10 0
• COllnlerciockwise • Clockwise
10 20
30
40
50
60
70
Time Slcp/ms
Time slep/ms
(a)
(b)
800 700 600 500 400 300 200 100 0
80
90
100
• COll nlerclockwise • Cloekwise
100
200
300
400
500
600
700
800
900
1000
Time slep/ms
(c)
Fig. 10.2
Testing curve of angular displacement of TRUM-45 vs. poweron time 60 E
'" ~ ~ <> «: "-a .';., ::
40
~ ,~
20
,~
~
:; '" 01) ~
• COllll lerclockwise . C loekwise
50
30
10 0
• I
3
5
10
20
• 30
50
100
200
500
Time Slcp/ms
Fig. 10.3 Testing curve on repeatability of angular displacement relating to poweron time for TRUM-15
The previous test shows that the USM's repeatability is bad III the startup stage. When the poweron time is more than 25ms during which the startup stage is completed, its repeatability of the displacement is acceptable. Thus, the time relating to the designed step angle should be far more than the time of the startup stage when the motor is used for stepping positioning. 1. Open-loop control method based on the poweron time as control variable The size of the step and the positioning precision are relied on the startup and shutdown characteristics of the USM. As a result the exact data of the starting
Chapter 10
Step Ultrasonic Motors
305
and the decaying time are required. from which one can determine the relationship bctwcen thc running time and thc displaccmcnt. Generally, there is a frequency automatic tracking technique for the USM. which can make operating speed stable. In the shutdown stage the slip friction mainly depends on the charactcristics of the frictional intcrfacc. which can inducc time-variability and uncertainty of the speed. Besides the problem of the slip friction, there are also other unstable factors such as the overshoot. However the time required for the whole stage of starting and ending is almost invariable. Hence, it is better that every time step includes the time for the starting or decaying stage to reduce the positioning error of every single step. A micro control unit (MeU) is a unit used for controlling stepping movement. The starting time and dccaying time and its corresponding angular displacemcnt at different loads are obtained through the cxperimcnt in advancc. Thcse data are stored in the memory of the MeU in table format. When the MeU receives the instruction of the step pitch and the numbcr of thc step. which will bc chcckcd out with starting, cnding timc and thc corrcsponding angular displacemcnt accorded in the tables. At a certain rotational speed. the MeU will give the steady running time for every stcp and the corrcsponding control signal, which will control the opcrating timc of a signal generator and a power amplifier so that the stepping motion of USM can be implemented. Single-step running is showed in Fig. 10.1. The time for USM's startup is t, , in which thc anglc that is rotatcd by thc motor is Similarly the dccaying timc is t" the angle rotated by the motor during ( is c • The angular speed and corre-
e,.
e
sponding running time in steady state are iJ and t p
•
respectively.
/I
lIs
Fig. 10.4
Fig. 10.5
USM's stepping motion
controlled by poweron time
USM's stepping motion
controlled by number of excitation periodicity
e
If the motor's displaccmcnt of thc ith step is step should be t,
i
thcn thc timc uscd for thc
+ tp + t,
e e, - e i -
c
iJ Thc powcron timc is
•
(10. 1)
306
Ultrasonic Motors Technologies and Ap plicalions
(10. 2)
The minimum interval time between these two steps is t,. The minimum step angle including a starting proeess and a deeaying process is It is relevant to the rotational speed. The higher the speed is, the larger the step angle will be. Taking TRUM-45 as an example, the minimum step angle is about 15 degrees in the clockwise direction while in the counter-clockwise direction is about 20 degrees including the entire starting and decaying stage, and its corresponding positioning error is 10 %.
e, + e,..
2. Open-loop control method based on periodicity number of excitation as control variable The open-loop control method based on the periodicity number of excitation as the control variable hopefully gets more position precise than that obtained by the previous open-loop control, and the differences mainly show in the process of shutdown. In the previous way, the driving signal hardly controls the motor to stop at the moment of the stator's vibration. This leads to the difference in the initial conditions when the stator starts the free vibration with damping. While nowadays, the exact initial conditions can be known. The similar initial conditions mean that the similar decaying time can be obtained. This controlling method is similar to the first one. The periodicity number of the excitation for startup of the USM and its corresponding angular displacement and the decaying time and the corresponding angular displacement in different loads are obtained by experiments. Furthermore, the data are stored in the memory of DDS (Direct Digital Synthesizer) in data tables. When DDS receives the instruction of the step pitch and the number of step, which will check out the periodicity number of the excitation for startup and decaying of the USM and its corresponding angular displacement accorded in the tables. From the angular displacement relating to the single periodicity number of the excitation, DDS will give the periodicity number of the excitation for every step when the USM operates steadily. The step movement can be controlled by a power amplifier. The periodicity of single step operating is shown in Fig. 10. 5. It is assumed that the periodicity number of the excitation for startup of USM is n,. During this period the rotational angle of the motor is e,. The shutdown time is t, and corresponding rotational angle is e" the stable angular speed is &and the operating frequency is foi' If the angular displacement of the ith step is i , the periodicity number of the excitation for this step is
e
(10. 3)
The minimal time interval between the two steps is tc' The minimal step angle
is
e, + e, including the starting stage and decaying stage.
Chapter 10
Step Ultrasonic Motors
307
3. Closed-loop control method using the position sensors for feedback The control prccision of thc opcn-Ioop control is primarily relatcd to the stability of the startup and decaying processes of USM. Because of the poor stability. the feedback must be added to reduce the single-step displacement error and the accumulated error brought by the multi-step operating. A relatively direct approach is to utilize a position feedback. Photo-electricity encoders or other position sensors can be used to acquire the location information. The informations acquired are then sent to the computers through the data acquisition card. The computer will cut off the driving signal in advance based on the built-in shutdown time tc of the motor. so that the motor will stop in the target location. The control precision of this method is not relevant to the startup process of the motor. but primarily depends on the stability of both the shutdown process and the resolution of the photo-electricity encoder. This method can get small step angle because it is not necessary to a complete startup process. However. it should be noticed that when the poweron time is less than startup time. the stator begins to enter the state of decaying from nonresonance of the stator with smaller amplitude and smaller speed. Different poweron time corresponds to different exciting state. The list of angular displacement relating to the different poweron time and poweroff time needs to be placed in the computer's memory in advance. The stepping control is shown in Fig. 10. 6. The shutdown time is t,. During this period the turning angle of the motor is B,. The angular speed of the motor is e(t). The computer's instruction execution time is Tafter computer gives the instruction of shutting down the power. :'\lamely. the computer will send the pow-
eroff command when the motor is at a location B = B, + TB (t) away from the target one. If the angular displacement of the ith step Bi is required and during the startup time t, the turning angle of the motor is B,. then the step angle Bi can be written as (10. 4)
e
From Eq. (10.4). Bi can be reduced by decreasing B,. B, and (t). and can reach a minimal step angle (B,> min. Previous three control methods can use a control strategy called DCM (Differential Composite Motion) to get a smaller step angle. Tsinghua University Xiangcheng Chu. Zengping Xing. et al. described this strategyllo-. and fulfilled the step control in a shaking head type USM. whose diameter. length and operating frequency are ¢15mm. 42mm and 30-45kHz. respectively. It's no-load speed is 150r/min. stall torque 0.12N·m. startup time o. 24ms. and minimum step angle 12" . The step angle Bi can be obtained by the way of turning forward and backward. As shown in Fig. 10. 7. the every step contains two processes: one is starting forward. running and decaying; the other is starting backward. running and decaying. There are some advantages in this way: the smaller step angle and greater output torque can be obtained. The displacements in the two startup and deca-
308
Ultrasonic Motors Technologies and Ap plicalions
ying processes can be subtracted with each other, which can reduce the error from instability statc. But thc way is forbiddcn whcn thc ovcrshot is not allowcd. Po it inof shutdown
~
80
~
60
§
'"
~
'is
TfJ( r) 0,
Fig. 10. 6 Stepping motion of USM using closed-loop control
10. 1. 3
40
O
~-L-L~L-~-L
500 I 000
__
I 500 2 000
I /ms
Fig. 10.7 DeM's locating strategy of stepping
Factors Impacting on Single-step Positioning Accuracy
For the step USM, the single-step displacement accuracy depends on its stability including the stability of stator's vibration and the force transferring to friction interface at different time and in different environment.
1. Stability of stator's vibration The main factor influencing the stability of stator's vibration is the change of connecting performance between the stator and piezoelectric ceramics. At present, there are three ways to connect a stator with a piezoelectric ceramics: compacting way. welding way and bonding way. (1) Compacting way Compacting way is conducted by a clamping force induced with connecting bolt, which makes both the modal frequency and amplitude sensitive to the clamping force. When USM is running, the heat from both the piezoelectric ceramic pieces and the friction interface will cause the stator's temperature rise. Due to thermal expansion the clamping force changes, then results in the fluctuation of the stator's modal frequencies and amplitudes. In addition, after long running, the motor's rated speed will change. Generally, thc piezoclcctric constant d" is bigger than its d 31 • and the clamping method is more suitable for these transducers that take the d" piczoelectric effcct, such as an ultrasonic motor using longitudinal-torsional bybrid vibration (scc Chap. 8). (2) Welding way Welding way is the mcthod that joins piczoclectric ccramic picces and a stator under a high tcmperature. which is bcneficial to more cffectively transfer piczoclectric ceramic deformation to the stator; further, it can provide high excitation cfficicncy along with dccreasing mcchanical hystcresis. In addition, the wclding possesses high rcliability and stability. and thc performances of USM arc preferable. However, in general, with the stator vibration. the strain of the piezoelectric ceramic picces on the welding surfacc and thc strain of the stator surface are
Chapter 10
Step Ultrasonic Motors
309
not totally consistent. This leads to a loealized stress concentration when USM is running, and the welding layer will has fatigue failure. (3) Bonding way Bonding way is the method of which water glue cement and so on bond a piezoelectric ceramic pieces to a stator in certain temperatures and pressures. Currently, an epoxy resin and acrylic adhesive is uscd extcnsivcly for USM. The adhcsivc possesses sufficient strength, which can provide thc clastic link bctween the stator and piezoelectric ceramic pieces and release the stress concentration on the bonding surface. Mostly this approach can be just utilized for the d 31 piezoelectric effect. Beside the thickness of the adhesive layer has great effect on the excitation efficiency, tempcrature changc also affect the clasticity of thc layer. Comparatively, the welding way and bonding way can make USM operating stable, while compacting way can induce some unstability.
2. Stability
0
f friction interface
Good performancc of thc friction interface can cnhancc the torque or thc running velocity of USM. A moderate friction coefficient, high hardness and high wear ability are thc basic rcquiremcnts for thc friction interface. Thc stability of thc friction interface is affcctcd by thc factors including the charactcristics of friction materials, thc surfacc morphology of the intcrfacc and the changes of the tribopair to cnvironmcnt, such as tempcraturc, humidity, and vacuum (scc Chap. 3). Thc stability of friction interfacc is a kcy to achieve thc stablc step interval of the step USM. The wear and tear of the tribopair change operating condition. From analysis above, we can draw the following conclusions: applying the materials with a moderate friction coefficient, high hardness and high wearability is bcneficial for obtaining more stablc friction interface.
10. 2
Step USM with Fixed Step length
10.2.1
Standing Wave USM Used for Constructing Step USM
The standing wave USM discussed in this section is a type of rotational USM whosc stator is ring shapc, as shown in Fig. 10.8. Thcre arc 4 tccth distributcd equally on the ring stator. Thc rotor is prcssed on it by thc prc-prcssure providcd with a spring. The monodirectional polarized piezoelectric ceramic ring has 8 uniform electrodes, as shown in the Fig. 10. 9. Figure 10. 10 is a diagram expanded in the circumferential direction of the USM. Thc figurc shows that thc stator's modc can be excitcd by thc driving voltage. This mode is the standing wavc ¢({}). In the coordinate of Fig. 10. 10, the standing wave can be expressed as: ¢({}) =
Wo
+
sin(wt Wo
+ ({J) cos(k{} -
sink{}sinkasin(wt
ka )
+ ({J)
= =
Wo
cosk{}coska sin(wt
¢l ({})
+ ¢, ({})
+ ({J) (10. 5)
Ultrasonic Motors Technologies and Ap plicalions
310
1
Pre-pressure
I'.
Fig. 10.9 Piezoelectric ceramic's subareas of uniform electrodes of rotary USM using standing wave
Fig. 10.8 Basic structure of rotation USM using standing wave
z
Stator mode __ a _ /
d
"'- I I
0
Fig. 10. 10
'"
I
"-
/1 1 ./
I
II "-
B
./
-Eosm (011) EUSlll
(rul)
Operating mode of standing wave type rotatory USM
=
0,
Mode ¢(B)
Mode I Mode 2
()
Fig. 10. 11
Where epj (B)
Decomposition of stator's operating mode
+
+
coskBcoska sin(wt rp), ep2 (8) = Wo sinkBsinka sin(wt rp) , k is the number of the standing wave in the stator, and here k = 2. ep] (B) and ep2 (B) are two standing waves whose phase difference is rr/2 decomposed from ep(B) in thc spacc. Moreovcr they are callcd as the mode 1 and mode 2, rcspecti vcly, as shown in Fig. 10. II. If tceth are takcn as rigid, tccth' s movcmen t will bc dctermined by thc rotational and translational movement of the points on the neutral layer of ring which is in thc bending vibration. In this figure, the tooth No.1 lies in the location of B = rr. The movement of the point on the neutral layer of the ring in the z direction is =
Wo
(10. 6)
The teeth's translational movement is determined by mode 1. That is to say, thc point's angle of rotation is
y
=
arctan ()ep(BB) I rd
a-IT
()ep2 (8) I arctan --Brd
a-IT
(10. 7)
Chapter 10
Step Ultrasonic Motors
311
Teeth's rotary movement is determined by mode 2. Where r is the stator's outer diameter. It is valid to assume that the distancc between one point on the top of tooth No.1 and the neutral layer of the ring is h, and then the movement of this point in the () direction is
~siny """ ~
{)t =
r
r
CJcp, (()) I rd{)
= O-IT
~ kw o sin(wt + rp) sinka
(10. 8)
r
Movement of this point in the z direction is Wo
coskCi' sin(wt
+ rp) (10. 9)
From the deduction above, the mode 1 makes the tip of the tooth to move in thc z direction, while the mode 2 makes the tip of thc tooth to movc in thc () direction. Further, Zj and {)j correspond to the point's amplitude of the tip of thc stator tooth in the Z and () direction, respectively. Giving the pre-pressure, an excitation voltage and height of teeth, the angle Ci' bccome thc kcy indicator determining thc synthesis of displaccments. As the angle Ci' changcs, the wavc shapc form cd by the tip's movcment is decided. The angle Ci' can be determined from design requirement so that the polarized pattern of piezoelectric ceramic ring can be determined. Thc wave shapes of CPt ({)) and cP, (()) as thc Ci' changcs arc shown in Fig. 10. 12. 2
-;(0) ---- ¢ ,(O)
2
-;(0) ---- ;,(0)
--- ;,(0)
--- ;,(0)
0 -\
-\
-~
-2
-0
0
(0) a ; ;I[/ 12
2
2 (b) a ; I[/3
4
2
- \
-20~--------------~ 2 --------------~4
(c) a
~ I[/4
Fig. 10. 12
When 0 ~
Ci'
~
-20~--------------~--------------~4
(d)
cr ~ I[ /6
Decomposition of standing wave with different a
re/1 , the amplitude of horizontal movement is larger than the
Ultrasonic Motors Technologies and Ap plicalions
312
one of vertical movement; when rr/ 1 ~ a ~ rr/2 ,the latter is larger. The shape of the locus is determined by h and a. When the designer needs bigger torque, it's better to make Zt larger. If the designer needs bigger speed, it's better to make at larger.
re=~=?f?? ~ ~ Jj=vsin(WI)~ ~/?74 ~ 3§s1l1«(~ L------(-a-)- - - - 1 - -E""" ,-:......:.J"'sl""' -; ·n"' (wc:./""")....
(a)
.
E- Vs in(w/)
Rotor direction
Rotor direction I (b)
(b)
Rotor direction
Rotor direction
I
(c)
(c)
Operating principle of rotor rotation in anticlockwise
Fig. 10. 13
Operating principle of rotor rotation in clockwise
Fig. 10. 14
Figure 10. 13(a) shows an operating shape produced by the annular plate type stator when a single phase voltage E excites the stator. Two of the four teeth of stator move in counterclockwise and contact with the rotor. The friction between the stator and rotor drives the rotor turning in counterclockwise. The other two teeth move in clockwise. Because they located close to the trough and do not contact with the rotor, then they cannot push the rotor. In the first half cycle of the stator's vibration. the teeth :'\10. 2 and No.1 contact with the rotor and drive the rotor, as shown in Fig. 10. 13(b). In the second half cycle of the stator's vibration, the teeth No.1 and No.3 contact with rotor and drive the rotor, as shown in Fig. 10. 13(c). Figure 10. 11 shows the driving voltage which makes the rotor turn reversely. The vibration mode is shown in Fig. 10. 11 (a). The moving decomposition of the teeth's tip and the rotor rotation are shown in Fig. 10. l1(b) and Fig. 10. 11(c).
10.2.2
Modal Rotary Type Step USM
1. Structure of USM The structure of a modal rotary type step USM is basically the same as the previous standing wave type USM. It is made of the annular stator, piezoelectric ceramic ring, spring and rotor. The difference is that the teeth arc on the rotor instead of on the stator, as shown in Fig. 10. 15.
2. Driving mechanism The stator has a mode
BOk
excited by the piezoelectric ceramic ring. According to
thin plate vibration theory, the displacement of point P on the stator's surface in
Chapter 10
Step Ultrasonic Motors
313
1
Pre· pressure Po
Spri ng Rotor
ROlDr Icelh
Fig. 10. 15 Basic structure of modal rotary step type USM and electrodes of piezoelectric ceramic ring
the z direction can be expressed as wp(r,{),t) = R(r)sin(k{))sin(wt)
(10. 10)
The speed of point P along the () direction can be expressed as (10.11) wherc d is thc thickncss of the platc, k is the numbcr of thc wavc, w is thc driving angular frequency, r is the radial coordinate of the point, and R(r) is the Bessel function
Whcre J n (kr) and Y n (k r) arc the first and sccond type nth ordcr Bessel functions, respectively; In (kr) and Kn (kr) are the first and second type nth order corrected Bcssel functions. rcs pecti vely. The rotor motion is acquired by two actions from the contact of the stator with rotor. One is the motion of the points on the stator's surface, and the other is the circumfercntial component forcc at the contact points of the stator with rotor. From Eq. (10. 11), the spccd V, of point P along () dircction, which locates at thc stator's surfacc bctwcen thc wavc crest and nodal diameter, changcs within thc cyele of the stator's vibration. as shown in Fig. 10. 16. In the first quartcr ofthc cyele of the stator's vibration (namely, wt 16a) or thc fourth quarter (i. e. wt
=
=
0-rr/2, as shown in Fig. 10.
3rr/2-2rr. as shown in Fig. 10. 16(d)), thc
speed V , direction aims at the nodal diameter of the mode shape. In the second quarter (i. e. wt=rr/2-rr, as shown in Fig. 10. 16(b)) or in the third quarter (i. e. wt=rr-3rr/2, as shown in Fig. 10. 16(c)), the specd V, direction aims at thc crests (or trough) of the mode shape. If there is no relative sliding on the contacting surface between the stator and rotor, the rotor will acquire the momentum in the circumferential direction that depends on thc contacting period bctwcen thc rotor's tccth and stator.
314
Ultrasonic Motors Technologies and Ap plicalions
z z
(WI =0- 1[/ 2)
Es in (w/) (w l = x - 3x/2) (c)
z
z
Es in(w!} (wl =3x / 2 - 21t )
Es in(w /) (w / =1[ / 2-x)
(d)
(b)
Fig. 10. 16
Speed component V, of contact point on stator
along circumferential direction
z
Es in (w l) (wl = 1[ -37t / 2) (e)
Esi n(wf) (w / =O-1[ / 2) (a)
z
Es in(WI) (wl =1[ / 2-1[) (b)
Fig. 10. 17
Esi n(W/) (WI =31[ / 2 - 2x ) (d)
Forcc component jd of contact point on
rotor along circumferential direction
The force fd in the circumferential direction is caused by the changing the slope on the contacting surface of the stator with rotor under the pre-pressure Po. The direction of the force fd also changes during the cycle of the stator's vibration, as shown in Fig. 10. 17. In the first quarter of the cycle of the stator's
Chapter 10
Step Ultrasonic Motors
315
vibration G. e. wt=O-rr/Z. as shown in Fig. 10. 17(a» or in the second quarter (i. e. wt= rr/Z-rr, as shown in Fig. 10. 17 (b», the force Id direction aims at the nodal diametcr of the mode shapc. In thc third quartcr (i. c. wt = rr-3 rr/ Z. as shown in Fig. 10. 17(c» or the fourth quarter of the cycle of the stator's vibration (i. e. wt=3rr/Z-Zrr. as shown in Fig. 10. 17(d». the direction of the force Id aims at thc trough or crests of the vibration shapc. If thcre is rclative sliding on the contacting surfacc between thc stator and rotor. thc rotor will be driven by the circumferential force component Id whose direction depends on the contacting pcriod betwecn the rotor's tecth and thc stator. If the rotor only contacts with thc stator during the first quartcr of thc cycle of the stator's vibration, which can assure that the motion component V, has the samc dircction with thc forcc component Id. In fact, if the numbcr of teeth on thc rotor is as twicc as that of the nodal diamctcrs, the stator will contact with the rotor in the first quarter or the second quarter of the vibration, i. e. wt= O-rr/ Z or wt=rr/Z-rr, as shown in Fig. 10. 18.
()
()
£s in Q)/ (Q)/"O- rr)
Es in (v / (Q)t=rr-2rr) (b)
(a)
Contact points between the stator and rotor
Fig. 10. 18
8
Fig. 10. 19
(01
Force analysis of rotor's teeth
Fig. 10. 20 Relationship of V, and Id vs. time
Assuming that the rotor contacts with stator during the first 1/1 cycle of the vibration, the circumfercntial component Id induced by contacting thc stator with rotor teeth under the pre-pressure Po can be expressed as
Id tanB
=
=
Po cosB sinB
1 dz(r,{),t) r
d{)
=
=
~o sinn
R(r)
k --cos(k{)sin(wt) r
316
Ultrasonic Motors Technologies and Ap plicalions
Considering that fj is small, there is an approximate relationship, as shown in Fig. 10. 19 sin2fj """ tan2fi
=
2 tanfj """ 2 tanfj 1 - tan'fj
Thus
fd """
Potanfj
=
R (r)
.
Pok --cos(k{})sm(wt) r
(l0. 12)
The changes of V, and fd versus time t are shown in Fig. 10. 20. In the period of the contact, the circumferential movement of the points on the stator's surface contributes to drive the rotor; while in the anaphase of the contact, the circumferential force component produced by contacting both the stator and rotor teeth under the pre-pressure will also drive the rotor more effectively. Thereby, the dcsign on the contact period of the stator with rotor is directly relatcd to the dcsign of frictional fcaturcs of thc contacting interfacc. The friction coefficient that is big enough can make the horizontal movement of the points on the stator to transmittcd reliably to thc rotor; the small friction coefficient makc thc stator and rotor teeth to possess relativc slide under the prcpressure, so that the circumferential force component Fd can come into better use for the driving force. It is impossible to design two friction materials with friction coefficients on the same surface, and only one of driving methods can be chosen to design the motor: (1) Usc horizontal movcment of the points on thc stator surface to drive thc rotor, which has large enough friction cocfficicnt. (2) Use the circumferential component force under pre-pressure between the stator and the rotor's teeth to drive the rotor, keeping the friction coefficient as small as possiblc.
3. Stepping principle Modal rotary type step USM relies on the vibration of the stator ring to make the rotor stepping. As shown in Fig. 10. 15, the piezoelectric ceramic ring is polarized in one way in the axial direction. The number of electrode is 20. Four teeth are distributed on the rotor uniformly, and B02 is the operating mode. On the purpose of convenient observation, the stator and the rotor are expanded along eireumferential. as shown in Fig. 10. 2l. The sinusoidal voltage is utilized for the electrodes of the piezoelectric ceramic ring, as shown in Fig. 10. 21(a). The bending standing wave Bo, of the stator is excited. The rotor's teeth move forward to the stator's nodal diameter under the effect of the stator, as shown in Figs. 10. 21(b) and (c). When the rotor's teeth locate at the nodal diameter, as shown in Fig. 10. 21 (d), the first step of the motor's movement will be over. When the step is finished, the power is supplied to the electrodes of the piezoelectric ceramic ring. From Fig. 10. 21(e) , the bending standing wave B02 of the stator is excited again, but at this time the standing wave has turned 21(/20 arcs around the motor's shaft. The rotor's teeth moves forward to the stator's nodal diameter, as shown in Figs. 10. 21 (0 and (g).
Chapter 10
Step Ultrasonic Motors
317
--r·L__ TT--TTT--TTT--IU-' ___ boSll1(<<>r) :
(a)
.
- 1'0
il1(",I)
Rotor direelion
Rotor direction
(f)
(b)
Rotor di rection 7777777777777T.;;;;;;;;~ ' '7T77777T77777T.
Fig. 10. 21
Rotor direction
(e)
(g)
(d)
(h)
Moving decomposition for modal rotary type step USM
When the rotor's teeth locate at the nodal diameter of the stator, as shown lil Fig. 10. 21(h). the second step of the motor's movement will be finished. Furthermore, changing connecting pattern of electrodes the power can make the standing wave turn step by step, and drive the rotor stepping, whose step angle is 2rc/20 arc.
4. Step number and step length In fact the minimum step angle of the motor is the minimum angle () which is the turning angle of the standing wave in the stator. The most step number of the rotor turning in one cyele is N = 2rc/{), which relies on the wave number k of the standing wave, the electrode number m", and the electrode number m, applied power. :'\J ext the mode BOk is discussed as an example to explain the relationships between these parameters.
If the operating mode of the stator is Bok
'
the wave number of the stator's cir-
cumferential wave is k and the wave length is A = 2rc/ k. The number of rotor's teeth is less than or equal to 2k. In order to make all points near each nodal diameter to drive the rotor, the number of the rotor teeth is 2k for the best. In order to excite effectively the bending mode. the angle of each sector area should be less than half of a wavelength, 1. e.
(l0. 13) The number of the electrodes should satisfy
2k
(l0. 11)
318
Ultrasonic Motors Technologies and Ap plicalions
If each crest or trough of the standing wave all can be excited by one corresponding sector area. it can be called as a "pure" mode exeitation. Here. the number of the group is me = 2k. and this approaeh requires the number of electrode is mp = 2jk (j is the natural number), which means mp is 2j times of the number k of the stator's circumferential wave. In such circumstances. the step angle is = 2rr/mp. As shown in Fig. 10. 22. B01 mode, the number of the group is m, = 8; the number of the eleetrodes is mp = 40; the step angle is = 2rr/40 and the number of the steps is N = mp = 10.
e
e
Fig. 10. 22
A "pure" mode excitation method
The smallest step angle using the" pure" mode exeitation method is only determined by the number of the sector areas (electrodes). Due to the restrictions of mechanical process. this approach will obtain a larger step angle as well as a smaller number of steps. The other excitation method is the non-pure mode excitation one, which means the number of group satisfies m, < 2k. In this way, the same step angle ean be received with less group. However, the number of group is reduced, and the problems arising from it is that the utilization for piezoelectric ceramics deelines and the output power is less. The relationship between the numbers of steps. the group and electrodes can be expressed as (l0.15) For the smallest step angle
e=
2rr m , =
rr m ,
2kmp
kmp
(l0.16)
An example is of the B01 mode: the number of group m, = 2. the number of electrodes mp = 10, the step angle = rr/20 and the number of steps N = 10.
e
5. Design of modal rotary type step USM Figure 10. 23 (a) is an assembly drawing of the 80-step motor. No. 01,02,03. and 04 are a rotor. rotor teeth. stator. and piezoeleetric ceramic ring, respectively. The outer diameter, inner diameter, and thickness of the stator are
Chapter 10
319
Step Ultrasonic Motors
60mm, 15mm, and 3mm, respectively. Fig. 10. 23(b) is the piezoelectric ceramic ring attached on the back of the stator, whose outer diameter, inner diameter, and thickness arc 60mm, 45mm, and O. 5mm, respectively.
(b)
(a)
Fig. 10.23
SO-step motor (a) and electrodes of piezoelectric ceramic ring (b)
A mplifier
Fivc·pha c ring cotllHer
Lock outpu t signal
D - r-iv-e -ig 'n- al:--------JJ f
'I
Fig. 10.24
Schematic diagram of five-phase drive circuit
The ceramic ring is evenly divided into 20 sector areas (namely m" = 20, which all are polarized in one direction). The sector areas are divided into 5 groups, which are marked the number 1, 2, 3, 1, and 5. Then each group has 1 sector areas, namely m,=4. The step number is N=80, and the step angle 11=271:/80. The step motor is known as the rotary step USM with five phases and 80 steps. Figure 10. 21 shows the element block diagram of the drive circuit for the rotary step USM with five phases and 80 steps. Fig. 10. 25 presents an actual driving circuit schematic. The five phases drive signals arc from a PWM controller TL494 (controlling the output voltage through adjusting pulse width), and the output frequency of the drive signal is elose to the resonant frequency of the stator. The five-phase ring counter is composed of two four-ring counters 10191. The stepping elock is controlled by chip 555. The output signal of the five-phase
~ ~:=t==
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Chapter 10
Step Ultrasonic Motors
321
ring counter through the gate circuit transmits the drive signals to the push-pull con trol circuit terminal. Figure 10. 26 shows the measured results of angle displacements when a rotary step USM with five phases and 80 steps is rotating in the two opposite directions. The maximum error of the step angle is about O. 7°. After running 800 steps. the ultimate angular error is no more than O. 7". In the case of load-free. the maximum speed of the step USM is 6. 25r/min. when the drive time of each step is not less than O. 2 second. The maximum output torque of the USM is O. 068N'm 45.0 ,...-..,-....,....--.,..---,-,....-..,-...,........,..---,----,.......,
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······i-· ..... j, .•••• j.. ' •••• ~. '·····l· ...... 5
6
7
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(a) Reverse rotation
(b) Forward rotation
Fig. 10.26 Step angular displacement measurement of a step USM with five phases and 80 steps at forward rotation or backward rotation
Fig. 10.27 Rotors and stators of mode rotary type step USM
Fig. 10. 28 Mode rotary type step USM using 5-phase driver
Figure 10. 27 shows the stators and rotors of the rotary type step USMs with a 80-step. a 120-step. and a 168-step[67: designed by the author. long Jin. et al. Fig. 10.28 shows a step USM using a 5-phase driver.
6. Features By modal rotation and changes of nodal diameter location. this kind of motors makes rotor teeth move to the nodal line and drives rotor to movement step by
11
322
Ultrasonic Motors Technologies and Ap plicalions
step. There is no accumulation of errors because it is from certain modes corresponding to the rotor position.
10.2.3
Self-correction Peak Type Step USM
A self-eorrection peak type step USM realizes its step motion by switehing its two operating states: standing wave driving and self-correetion positioning.
1. Structure and driving mechanism of the motor The structure of self-correction peak type step USM is similar to the aforementioned standing wave type one. It is made of a stator, piezoelectric ceramic ring, spring and rotor. The difference is that the rotor has some radial slots, as shown in Fig. 10. 29.
t
Pre-pressure
I'.
Spring
Fig. 10. 29
Self-correction (tuning peaks) type step
USM's structure (left) and electrodes (right)
(a)
Fig. 10.30
(b)
Self-correction mechanism of self-correction peak type step USM
This type of step motors uses the stator's bending mode as an operating mode, and its two operating states are determined by the relative position between stator teeth and the standing wave. When stator teeth are between the standing wave peaks (or trough) and the nodal diameter, the motor is in the state of the standing wave driving, and its driving mechanism is the same as the aforementioned standing wave motor. Meanwhile the friction at the interface is active. When stator teeth locate near the wave peak (or trough) , it is in the self-correction positioning state. Self-correction mechanism is shown in Fig. 10. 30. Under the action of pre-
Chapter 10
Step Ultrasonic Motors
323
pressure F n , the torsional deformation of the rotor's lobule leads to a eircumfercntial force F" as shown in Fig. 10. 30 (a), which drives the rotor rotating. Whcn stator teeth arc in thc rotor slot, thc circumfcrential forces F, and F', act at the adjacent two lobules of the rotor, which possess the same magnitude but the opposite directions, and the rotor is no longer rotating to realize the rotor's positioning. Thc relative sliding during thc contact of the stator with the rotor is a nccessary condition for the rotor's rotation. As a result, thc friction force at thc interface is resistance. Because stator teeth stand in the peak (or trough) is the self-correction state, this type of motors is called as thc self-correction pcak typc step USM. Thc torque in the self-correction statc dcpends on the torsional anglc, which is formed by the torsional deformation of the rotor's lobule turning around thc geometric symmetry axis under thc action of the stator tooth.
2. Stepping principle As shown in Fig. piezoeleetrie ring motor structure, valid to expand Fig. 10. 31.
10. 29, the step motor with 4 s ta tor teeth, 8 rotor slots and a unidireetional polarized with 12 eleetrodes are designed. In the B02 operating mode is utilized. To facilitate observation, it is the motor and rotor along circumference, as shown in
I I:.fJ (a)
m( rtJ /)
i
-b'osin(tti ')
(d)
-H.sin(tti ')
Response of rotor
•
"
./~""-....n......'l"~
-' •
~. (e)
Response of rOlor
~~eZ~2!eZ2&:
/-"-~ (c)
Fig. 10.31
-~ (f)
Motion decomposition of self-correction peaks type step USM
The voltage supply is applied to the electrodes in accordance with Fig. 10. 31 (a). The E02 bending standing wave is excited by the piezoelectric ceramic ring. Under the action of stator teeth, the rotor moves along the direction as shown in Fig. 10. 31(b) and Fig. 10. 31(c). While the rotor slot is in the vicinity of stator teeth, the voltage supply is applied to the electrodes in accordance with Fig. 10. 31(d). The E02 bending standing wave is excited by the piezoelectric ceramic ring, and at this moment stator teeth are in the place of peaks (or trough). The rotor's lobules are distorted by the effect of stator teeth, which contributes to a circumferential force to make the rotor slot finally to stop just in stator teeth, as shown in Fig. 10. 31(e) and 10. 31(f). It is the completion of self-correcting posi-
324
Ultrasonic Motors Technologies and Ap plicalions
tioning, and the motor's first-step movement has finished at the same time. In this way, alternately changing the electrodes in accordance with Figs. 10. 31(a) and (d), the rotor can alternately obtain the both operating states: the standing wave driving and self-correction positioning. Further, the motor step angle is
2 rr/ 8. The step angle of this type of step USMs equals to the angle between the adjacent slots on the rotor, and this is related to the number of rotor's slots. The more the slots are, the smaller the step angle is. The number of stator teeth is related to the usc of the operating mode. if we adopt the mode EOk , the number of teeth should be 2k. Fig. 10. 32 shows the self-correction peak type step USM developed by Chunsheng Zhao and Guiqin Wang L1 -S- . If using B12 vibration mode, the stator is installed with four teeth, and the stator's outer diameter is 10mm, the inner one is 21mm. the height of teeth is 3mm. the stator's thickness is 1. 5mm, and its material is 40Cr steel. The rotor is cut into a series of the lobules from the stepping angle, as shown in Fig. 10. 32. As shown in Fig. 10. 33, it illustrates the mechanical characteristics of the three kinds of self-correction peak type step USMs (as single-phase driving motors) at the same pre-pressure. Their driving frequency and driving voltage arc 32. 78kHz and SOV, respectively. 450.---~~--~-----r----~----,
,
.::
~
:a
....
300
8 slot copper rotor ---- 16 slot aluminum rotor
", ,
,
~ ~
"
_. - ,. 8 slot aluminum rOtOr
\
\
.\
...\.
' .\
,~ 150 <5
,. ".
~~
c<
00
0.005
0.010
0.015
0.020
0.025
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Fig. 10. 32 D1 0 self-correction peak type step ultrasonic motor
Fig. 10.33
Load characteristic curve
The relevant size parameters are as follows: 8-s10t copper rotor's thickness is 1. 8mm; 16 slot aluminum rotor's thickness is 3. Omm; a 8-s10t aluminum rotor's thickness is 3. Omm. Comparing the characteristic curves, the performance of the USM with a copper rotor is better, mainly because the copper's friction coefficient is larger and machining precision is higher.
3. Features As the friction at the interface in the two operating states produces the contrary effect, a suitable friction coefficient and a suitable pre-pressure arc main factors for operating of the motor, but it is very difficulty to increase a torque in integrated design. In addition, there is another key factor altering the states from the driving state to the positioning state through a switch. Too early switching
Chapter 10
Step Ultrasonic Motors
325
will make the rotor to baek to original position. On the contrary, switching too la te will make the rotor to enter in to afterward self-correction region, resulting in multi-step phenomenon. Corresponding with the self-correction peak step USM, there IS another selfcorrection step USM called the self-correction nodal-line type step USML36_, which also uses the mutual switch of standing wave driving and self-correction positioning to achieve stepping movement. The only difference is that its stator teeth locate at the nodal diameters of operating mode at the self-correction state.
References [ 1J
[ 2J
C Kusakabe, Y Tomikawa, T Takano, et al. An encoder-less ultrasonic stepping motor using open-loop control system. Proceedings of 12th Symposium on Ultrasonic Electronics. Tokyo: Japanese Journal of Applied Physics, 1992, 31(S): 239-241. T Iijima. Ultrasonic motor using flexual standing wave. Japanese] ournal of Applied Physics, 1987,23 (S): 191-193.
[3 J [ 4J
0 Miyazawa. Drive control unit for an ultrasonic step motor. US Patent 5229678, 199307-20. Chunsheng Zhao, Guiqin Wang. Self correction ultrasonic motor using standing wave. Chinese Invention Patent 1133265, 2003-12-31.
[ 5J
[ 6J [ 7J [8 J [ 9J
Guiqin Wang. Research on Self Correction Ultrasonic Motor Using Standing Wave. Dissertation for the Degree of Master. "fanjing: "fanjing University of Aeronautics and Astronautics, 1999. (in Chinese) Chunsheng Zhao, Long Jin. Multiphase stepping ultrasonic motor. Chinese Invention Patent 1299180, 2001-06-13. (in Chinese) Long Jin. Research on Stepping Ultrasonic Motors. Post-doctoral Report. Nanjing: Southeast University, 1999. (in Chinese) V Snitka. Ultrasonic actuators for nanometer positioning. Ultrasonics, 2000, 38: 20-25. Yong Jin, Jifeng Guo, Kehui Ji. Stepping positioning control on ultrasonic motor. Mechanical and Electrical Engineering, 2003, 20 (6): 25-29. (in Chinese)
[lOJ
Xiangehen Chu, Zengping Xing, Longtu Li, et al. High resolution miniaturized stepper ultrasonic motor using differential composite motion. Ultrasonics, 2001, 11: 737-711.
[l1J
Jiamei Jin. Development on Some Novel Ultrasonic Motor. Dissertation for the Degree of Doctor. Nanjing: Nanjing University of Aeronautics and Astronautics, 2007: 71-77. Chunsheng Zhao, Jiamei Jin. Frequency stepping ultrasonic motor using standing wave. Chinese Patent 1688097, 2005-10-26. Jiamei Jin, Chunsheng Zhao. Bi-mode alternation stepping ultrasonic motors. Journal of Nanjing University of Aeronautics and Astronautics, 2006, 38(5): 600-601. (in Chinese) Chunsheng Zhao, Jiamei Jin. Linear stepping ultrasonic motor. Chinese Invention Patent 1777011, 2006-07-10. Jiamei Jin, Chunsheng Zhao. A vibrators alternation stepping ultrasonic motor. Ultrasonics, 2006, 44(1): 561-564.
[l2J [l3J [14J [15J [l6J
Chunsheng Zhao. Ultrasonic motor techniques in the 21" century. Journal of Vibration, Measurement & Diagnosis, 2000,200): 7-11. (in Chinese)
[17J
Huafeng Li, Chunsheng Zhao. Precise position control of ultrasonic motor using fuzzy conIEEE Internatiunal [lltrasonics, Ferruelectrics, and Frequency Control ] uint 50 th Anniversary Conference. Montreal, Canada: IEEE, 2004: 23-27. Qianwei Chen, Weiqing Huang, Chunsheng Zhao. Measurement of service life of ultrasonic motors. Journal of Vibration, Measurement & Diagnosis, 2004, 24 (l): 19-22. (in Chinese) trol.
[18J
326 [19J
[20J
[21J [22J [23J
Ultrasonic Motors Technologies and Ap plicalions Xiangdong Zhao, Bo Chen, Chunsheng Zhao. Characteristics estimation and optimal design o[ traveling wave type ultrasonic motor. Small & Special Electrical Machines, 2003 (5): 1315, 19. (in Chinese) Shoushui Wei, Yuling Zhang, Chunsheng Zhao. The application o[ [uzzy control to USM-actuated position servo system. Electric Machines and Control, 2002, 6(3): 218-220. (in Chinese) Shoushui Wei, Hui Guo, Chunsheng Zhao. It novel method o[ phase control [or ultrasonic motor. Journal of Southeast University, 1999, 29(5B): 76-79. (in Chinese) liakui Zu, Chunsheng Zhao. Development of driving and control technique for traveling wave ultrasonic motors. Small & Special Electrical Machines, 2001(6): 38-12. (in Chinese) Zhirong Li, Weiqing Huang, Chunsheng Zhao. Motion analysis and simulation o[ the stator driving sur[ace o[ the cylinder sphere ultrasonic motor with multi-degree o[ [reedom. Mechanical Science and Technology, 2001, 23(11): 1352-1355. (in Chinese)
[24J
[25J [26J [27J
Hai Xu, Weiqing Huang, Chunsheng Zhao. One type o[ linear ultrasonic motor based on the vibration in plane of the plate. Journal of Vibration Engineering, 2003, 16(S): 38-10. (in Chinese) Quanting Liu, Weiqing Huang, Chunsheng Zhao. Linear ultrasonic motor and it's type applications. Journal of Vibration Engineering, 2003, 16 (S): 25-28. Wciqing Huang, Guoqing Huang, Chunshcng Zhao. Research on multi-stator linear ultrasonic motor. Piezoelectrics & Acoustooptics, 2001, 26(6): 151-153,159. (in Chinese) Hai Xu, Chunsheng Zhao. Development and application o[ linear ultrasonic motors. China Mechanical Engineering, 2003,14(8): 715-717. (in Chinese)
[28J
Heling Su, Chunshcng Zhao. Study on the dynamic modeling and simulation of a rotatory type ultrasonic motor with single phase driving circuit using standing wave. Chinese Journal of Applied Mechanics, 2003, 20(2): 78-82.
[29J
lunbiao Liu, Weiqing Huang, Chunsheng Zhao. It new type linear ultrasonic motor with two degrees of freedom. Piezoelectrics & Acoustooptics, 2001, 23(5): 316-358. (in Chinese)
[30J
Xiaohong Yuan, lunbiao Liu, Chunsheng Zhao. Study on a new linear ultrasonic motor with standing wave. Piezoelectrics &. Acuustooptics, 2001, 23(3): 198-201. (in Chinese)
[31J
Heming Sun, Shoushui Wei, Chunshcng Zhao. Research on ultrasonic motor using longitudinal and torsional mode. Journal of Nanjing University of Aeronautics & Astronautics, 2000, 32(5): 603-607. (in Chinese)
[32J
Chunsheng Zhao, Guiqin Wang, Long lin. It new type o[ self-correction ultrasonic motor using standing wave. IEEE Ultrasonic Symposium ,1999 ,2: 671-671. Chaodong Li, Chunshcng Zhao. A large trust linear ultrasonic motor using longitudinal and flexural modes o[ rod-shaped transducer. Proceedings of the 1998 IEEE International Ultra-
[33J
sonics Symposium. Sendai Miyagi, lapan: IEEE, 1998: 691-694.
[31J
[35J
Chaodong Li, Long lin, Chunsheng Zhao. The characteristics of hybrid transducer type linear ultrasonic motor with large thrust and large stroke. Chinese Journal of Acoustic s, 1999, 18(3): 266-271. Tiemin Zhang, Chunsheng Zhao. Comments on servo control over ultrasonic motors. Jour& Diagnosis, 2001, 21 (3): 203-208. (in Chinese) X Chen, C Kusakabe, Y Tomikawa, ct at. Rotor displacement of the ultrasonic motor having
nal of Vibration, Measurement
[36J
an angular displacement self-correction [unction. 1993,32 (1): 4198-4201. [37J
Japanese Journal of Applied Physics,
Chunshcng Zhao. Ultrasonic Motors Technologies and Applications. Beijing: Science Press, 2007. (in Chinese)
Chapter 11
Other Ultrasonic Motors There are many types of motors based on the piezoelectric effect (also called piezoelectric motors). Besides those discussed in previous chapters, there are many other new types being studied, including a non-contact type ultrasonic motor, a piezoelectric motor using clutch, a linear surface acoustic wave motor, a vacuum type ultrasonic motor, an impact type ultrasonic motor and so on. Since these types are still in the process of research and development, and many problems are not clear, the non-contact type USM and linear surface acoustic wave motor are mainly discussed here. For further studying of remaining types, interested readers can consult Refs. [1-,1].
11. 1
Non-Contact Type Ultrasonic Motors
The non-contact type ultrasonic motors were put forward and first studied by Japanese researchers in the 1990s[5 8:. While the ultrasonic motors mentioned before are driven by the friction between the stator and rotor, which restricts the increase of motor performance and life span, researchers started to try non-contact way to avoid friction and bring forward a new non-contact type ultrasonic motor. The driving mechanism of the contact type ultrasonic motor can be summarized by solid to solid way, that is, when traveling waves are excited in the stator, the points on the stator upper surface move in an elliptical fashion, and consequently the rotor placed on the top of the stator with pressure would be rotated in a direction retrograde to the direction of the traveling wave because of the friction, as shown in Fig. 11. 1. Compared to the contact type, the non-contact type ultrasonic motor uses solid to fluid to solid way, which transfers torque through fluid in the gap between the stator and rotor and causes the rotor rotate in the same direction as the traveling wave as shown in Fig. 11. 2. Compared with contact type ultrasonic motors, the non-contact type ultrasonic motor has the following features: (1) there is no contact between the stator and rotor. As a result, the loss caused by friction is avoided, and there is no need of friction material, life span becomes longer and structure simpler, but there is hardly holding ability; (2) the rotor can reach a very high velocity, because it is out of the restriction of stator; (3) the rotor rotates in the same direction as the traveling wave, and its torque is comparatively small.
C. Zhao, Ultrasonic Motors © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011
Ultrasonic Motors Technologies and Ap plicalions
328
Motion direction of rotor Pre-pressure Motion direction of rotor Rotor
Rotor
Direction of travelling wave ,
Stator
Fig. 11. 1 Driving mechanism of contact typc ultrasonic motor
Stator
Driving mechanism of non-contact typc ultrasonic motor
Fig. 11.2
In a word, the non-contact type ultrasonic motor has different performance from the contact type ultrasonic motor and its features of small torque and high velocity enlarge the application area of the ultrasonic motor, as shown in Fig. 11. 3. Application are a of non-contaclly pe ult rasonic
Application area of conlacllype US M Application area ormolor
o~------------------------· Torq ue
Fig. 11. 3
11. 1. 1
Applica tion area of ultrasonic motors
Classification and Development
At present, there is no unitive classification for the non-contact type ultrasonic motors. According to its shape, it can be classified into disk, cylinder and flat types. According to the driving medium used, it can be classified into liquid and gas types. According to its source of driving force, it can be classified into acoustic radiation pressure type, acoustic fluid type and mixture type of acoustic radiation pressure and acoustic adhesive force. The non-contact type ultrasonic motor only has the history of no more than twenty years and researchers studying it are mainly in Japan. In 1996, Nakamura, lunhui Hu, and their colleagues of Tokyo Institute of Technology designed a non-contact type ultrasonic motor made of aluminium with a maximum velocity 4 400r/min as shown in Fig. 11. 4(a), whose gap between the stator and rotor is o. 05mm and in which a traveling wave in stator is excited by two Langevin vibratorsL9-10J. Fig. 11. 1 (b) shows a non-contact type ultrasonic motor designed by Nakamura and his colleagues[5]. The stator is a hollow cylinder with the length of 20mm, inner radius of 50mm, and outer radius of 60mm. With ring-shaped
Chapter 11
Other Ultrasonic Motors
329
piezoelectric ceramics bonded onto the upper surface of the stator. the traveling wavc of modc E01 is excited along thc circumfcrcntial dircction of thc stator. Thc lower surfacc of thc stator is bonded onto a rubber basco Fluid such as water or kerosene is used as the acoustic medium contained in the cylinder. The maximum rotation speed of the motor is about 50r/min.
PZT
Langevin vibrator
,
Stator Air Rotor
Base (a) Excited by
Fig. 11. 4
twO
Lan gevin vibrators
Rotor
(b) Excined by ring-shaped PZT piece
Non-contact type ultrasonic motor made by Tokyo Institute of Technology Pi ezoelectric cerrul1 ics
(a) Stmcture of stator
Fig. 11. 5
(b) Struc.ture of rotor
Non-contact type ultrasonic motor made by Suzuki
Figure 11. 5 shows a non-contact type ultrasonic motor made by Suzuki of Yamagata U niversityL6J. The cylinder shaped stator with two pairs of piezoelectric ccramics bondcd on its outer surfacc is madc of stainlcss stcel, and has thc length of 90mm, inner radius of 10mm and outer radius of 12mm. The rotor is madc of thin cardboard. Whcn a traveling wavc in thc acoustic medium is synthesized by two orthogonal B02 modes. which are excited by two pairs of piezoelectric ccramics conncctcd to signals sinwt and cOSwt with the frcquency of 20. 5kHz, respectively, the motor can reach the maximum velocity of 950r/min. Figure 11. 6 shows a non-contact typc ultrasonic motor madc by Hirose of Yamagata University[7:. There is a disk shaped piezoelectric ceramic piece on the stator, an acoustic reflection plate is installed to strengthen the sound field. The rotor is madc of papcr. The gap bctwcen the stator and rotor is o. 7mm. When thc traveling wavc of E03 mode is excited through forkcd elcctrodc on the stator surface, the motor can reach the maximum velocity of 3 OOOr/min. The non-contact type ultrasonic motor as shown in Fig. 11. 7 designed by Japanese researcher
Ultrasonic Motors Technologies and Ap plicalions
330
Hanada also has a ring shaped piezoelectric ceramic stator and an airscrew rotor 8J driven by the acoustic radiation pressure induced by the water contained in the cylinder container. The maximum velocity of the motor is 800r/min.
®
Reflecting plme
---Bearing
~=~
-Cylinder
Rotor
~ -Acoustic absorber 30l11lll --~~:-II----+- Renccling
}-=:::::::.d ~Z?if 0. 7111 111
fool.---::-::---~~
Fig. 11. 6
plale
ROlor Slator
Non-contact typc ultrasonic
motor madc by Hirose
~
- - - Airscrew rotor
~-Rubber ~ Fig. 11. 7
-Piezoelectric ring
Non-contact type ultrasonic
motor made by Hanada
Figure 11. 8 shows a non-contact typc ultrasonic motor madc by Quanjing Liu, Zhigang Yang, and their colleagues- llJ . Its cylindrical stator equably bonded with four rectangular piezoelectric ceramic pieces is made of duralumin, and has the length of 10mm, inner radius of 36mm and outer radius of 10mm. When the traveling wave made of of B23 modes is excited by four voltage signals sinwt, coswt, - sinwt, and - cOSwt with thc frcqucncy of 23. 4kHz applicd to four piczoelcctric ccramic picccs, rcspcctively, thc cross-shapcd rotor will rotatc in thc samc dircction as thc traveling wavc, and rcach thc max. velocity of 670r/min. Using thc samc structurc as Fig. 11. 8, Zhigang Yang madc a non-contact typc ultrasonic motor drivcn by singlc phasc voltagc signal. With two piczoelcctric cc-
[Stg--
Rotor
~ pZTri"g
~
---7-t-- Fluid
Fig. 11.8
:-..ron-contact type ultrasonic
motor made by Jilin University
Fig. 11.9
:-..ron-contact type ultrasonic
motor made by Tianjin University
Chapter 11
Other Ultrasonic Motors
331
ramie pieces equably bonded onto the outer surface. the cylindrical stator will vibrate in mode B03 when the voltage signals sinwt and - sinwt with the frequency of 19. 95kHz are imposed on the two piezoelectric ceramic pieces. respectively. The motor has the maximum velocity of 2 026r/min and thc torque of 184 X 10- 6 N· m. Fig. 11. 9 shows a non-contact type ultrasonic motor made by Chang liang Xia of Tianjin University-12J.
11. 1. 2
Operating Principle 10]
There arc many non-lincar phenomcna
III
acoustics such as acoustic radiation
pressurc. acoustic strcaming. acoustic adhesive forcc ctc., which arc used as source of driving force of a non-contact type ultrasonic motor. These phenomena will be discussed in detail as follows.
1. Acoustic radiation pressure Due to the special non-uniformity of energy density around an object in acoustic field. a unidirectional pressure with fixed direction and magnitude which is called acoustic radiation pressure will be produced. as shown in Fig. 11. 10. Although acoustic radiation pressure phenomenon was discovered early in the 19th century. its research on theoretical explanation was not obtained until Rayleigh studied it in the beginning of the 20th century.
From that on. many
researchers have been researching on it. and only two different theoretical results have been achieved.
Theoretical formula derived by Rayleigh Chiefly. Chu.
Apfel, et at. is in Ref. [13 ] (1l.1) where y is specific heat capacity of air and (E) is the average energy density of sound field. Langevin Chiefly. Hetz. Niberge. et at. derived following formula: p
R
=y+1(E) 2
(ll. 2)
which is two times of Eq. (ll. 1). At present. there is still no unitive theory about acoustic radiation pressure. Based on Eq. (ll. 1). Japanese researcher U eha and Hashinmoto studied acoustic radiation pressure in research on the non-contact type ultrasonic motor and derived an approximate formula as Eq. (11. 3). which can be applied to estimating the acoustic radiation pressure in non-contact type ultrasonic motor and the result was proved true by experiment111 -. p
= R
y+1(1+ sin (2ka»)( P: +PoV2) 2 2ka 2po c 2
(1l.3)
where c is the sound speed. k = w/ c is the wavenumber. w is the angle frequency of sound wave. a is the gap between the vibrator and object suspended. v is the vibrator velocity and" - " denotes the average value of time during several cyeles. When gap is small enough to fulfil ka 1. Eq. (ll. 3) can be approximately
«
Ultrasonic Motors Technologies and Ap plicalions
332
written as
PR
=
l+r
-4-Poc
,A'
(ll. 1)
(J'
in which A is maximum displaccment of vibrator surface. According to approximate Eq. (11. 1), the change of the acoustic radiation pressure along with gap can be calculated, as shown in Fig. 11. 11. From the figure, acoustic radiation pressure becomes bigger with the increase of amplitude, and becomes smaller rapidly with the increase of gap. When the gap is equal to the amplitude, acoustic radiation pressure reaches maximum value. It is not in the levitation state when the gap is less than the amplitude, for which the acoustic radiation pressure formula does not hold. If the influence of surface roughness is taken into account, the maximum value of acoustic radiation pressure will decrease greatly. For example, provided both surface roughness of the stator and rotor of non-con tact type ultrasonic motor is
o. S,um,
the maximum theoretical
acoustic radiation pressure will be 74 082N/ m', which is bigger than the value 70 413='J/m' measured by Japanese researchers through experiment. The difference is mainly because of surface roughness.
Vibrator Ultrasoni c w ave Acoustic radiation pressure
Fill id
Sketch map of acoustic radiation pressure
Fig. 11. 10
1 ~~ O-~------~ I O~-'-------~ I O~-1 Gap/ mill
Relationship between acoustic radiation pressure and gap
Fig. 11. 11
As diseussed above, a vibrating stator with high frequency can induce a high acoustic radiation pressure on the rotor near the vibrating sur face.
For in-
stanee, the cylindrical non-eontact type ultrasonie motor designed by PDLab, which can reach the maximum radiation pressure of 206='J, has the radiation pressure of 18. S='J when the gap between the stator and rotor is
o. Smm
and the
amplitude of the stator is S,urn. 2. Acoustic streaming When an ultrasonic wave travels through gas or liquid, a kind of non periodical
motion of fluid called an acoustic streaming will be generated.
The acoustic
streaming phenomenon, which can be classified as boundary layer type and far from boundary type, is rather complex. The minority of non-contact USMs was driven by the acoustic streaming far from boundary, as shown in Fig. 11. 12, that
Chapter 11 IS,
Other Ultrasonic Motors
333
to make the fluid far from the stator, to flow macroscopically through the vi-
bration of stator, and thus to drivc thc rotor to move togethcr with the fluid.
3. Acoustic adhesive force Most non-contact USMs are driven by boundary layer acoustic streaming, which lies on surface of an object in strong sound field, as shown in Fig. 11. 13. Likewise, the stator of non-contact USMs vibrating with high frequcncy crcates strong sound field and thc boundary laycr acoustic streaming is gencrated on thc surfacc of a rotor. Bccausc of the fluid viscosity, acoustic strcaming will act on the rotor as a tangential force called acoustic adhesive force, which is the driving source of non-contact USMs. Z
Vibrator _ _ _•
Acoustic field
Boundary layer aco ust ic stream
Ultrasonic wave Aco ustic stream ing
Boundary layer thickness
o
Fluid
Fig. 11. 12
- - - - -., Acoustic adhesive force
Sketch of acoustic
Fig. 11. 13
Sketch of acoustic adhesive force
streaming
8
0.45 ,---------,----,,-----,---,,.....,------,-----.----,
~ 0.4
7
'?
~0.35
"~
0.3
~5
§ 0.25
<>
g.
1;1" 0.2 .~ 0.15 f--f--+<"--Bounday layer
8 0.1
<;
,.,
kHz _. _ . _ 1/= 1
f~ 25
./
- - 11- 3 - - - - /F2
.'
4
2i
.'
2
"
I
.'.
.--
"
O ~~~~~=±~~~~
2
3 Gap/m
4
Acoustic stream speed vs. gap
Fig. 11. 14
10- 6
;. 3 .~
thickness
< 0.05 00
6
X
6
0.05 0 I 0. 15 0.2 0.25 0.3 0.3 5 0 .4 0.45 0.5 Vibrati ng speed of stator/(m / s)
Driving torque vs. vibrating speed of stator
Fig. 11. 15
Based on the analysis methods for the boundary layer acoustic streaming and acoustic adhesivc forcc prcsented by ='Jyborg, Lec and Wang, thc author and Yc Ji sct up a non-contact driving model for his non-contact USM with a cylinder stator· l5 · l7 J. Limitcd by lcngth, this book docs not quotc the derivation and results because its derivation and formulae are complicated, and it is hard to find obvious relationship between adhcsivc forcc acting on the rotor or driving torquc
Ultrasonic Motors Technologies and Ap plicalions
334
and the vibration velocity of the stator or the gap between the stator and rotor. Figs. 11. 11-11. 17 show simulation results based on the driving model. Fig. 11. 14 shows the relationship between the acoustic streaming speed and the gap, in which the broken line stands for the outer boundary of the boundary layer which is so thin that there is usually only a few ten microns in high frequency sound field. Although the acoustic streaming speed is not very high in the boundary layer ncar rotor, the gradient of the speed is quite large ncar the boundary. This causes a remarkable tangential adhesive force. It can be seen from Fig. 11. 15 that the driving torque of the non-contact USM with the order of magnitude from 10- 6 to 10- 5 N· m is far lower than the contact type USM when the maximum vibrating speed of the stator is in the range of O. os to O. Sm/ s. The torque becomes larger when the vibrating speed or the order of mode in stator is increased. This nonlinear characteristic can be used to adjust the speed of the motor, that is, the adjustment of the speed of the motor can be conducted with changing the vibrating speed or operating mode of the stator. Figure 11. 16 shows that the gap has a remarkable influence on driving torque. The smaller the gap is, the bigger the torque becomes, which is an important relationship for the motor design. To increase the driving torque, the gap should be as small as possible. According to the application condition, the design of gap should also consider some other factors such as asymmetry of gap or rotor's eccentricity caused by machining, which will result in partial point contact between the stator and rotor, leading to the decrease of the speed, even stop and reversal rotation of the motor. In a word, all kinds of factors should be taken into account with the gap. Fig. 11. 17 shows the driving torque versus frequency under certain vibrating speed. The peak of driving torque locates near 18kHz, before which the torque increases rapidly with frequency and after which it decreases slowly. x IO- S 3r---~---'----~--~----'
2.5
E
2
~ "5- 1.5
2
1 - 20 kHz
:
2.5
11 - 3 E
Z
". Vibrating speed of stator
....... / '.,
"'S!0-"
,,- 0.5111/S ,F(). 4111/s
OJ)
~. 3
<= .;:
·c Cl
X
2
0. 1
2
IF 3
mill 1'- 0.5 ",Is
cr~. 1
1.5
011
:~
m1s
£5 0.5
0.5 0 0.05
10-"
3,..----r---,..----r---r----,
0.2
0.25
Gap between rotor and tsatorlmlll
Fig. 11. 16 Driving torque vs. the gap
0.5
0 '----'-----'---'-----'-----' 1.5 2 2.5 3 3.5 4 1 1Hz x 10'
Fig. 11. 17 Torque vs. frequency under fixed vibrating speed
Based on the driving model set up by author and Ye Ji, from its simulation results, we made two kinds of non-contact USMs with cylinder stator and disk
Chaptcr 11
Othcr Ultrasonic Motors
335
stator-18-19J , respectively. Their design and test research method-ZO- 22J will be discusscd as follows.
11.1.3
Design of Non-contact USM
1. Design of cylinder stator Thc stator is a hollow cylinder which operates in the B= bending vibration modes according to vibration analysis in Chap. 2. where n is the number of circumferential waves, m is the number of the nodal sections along the axis. Since it is hard to place piezoelectric ceramic pieces due to the nodal cireles along the axis when m =F 0, comparatively simple Bon bending modes were chosen as operating modes. It can be seen from Fig. 11. 15 that the larger n is, the higher torque will be. However, for too many circumferential waves it will be hard to bond piezoelectric ceramic pieces to the outer surface of cylinder, but many piezoelectric ceramIC pIeces are needed. and the difficulty of machining and assemblage will be increased. Here Boo is chosen as the operating mode, as shown in Fig. 11. 18.
sin"'I:.....t_ +~.....PZT Stator
+
+
- cos(U/
+ - sin«'1 Mode cy linder stator
Fig. 11. 18
B03
of
Arrangement of PZT on cylindcr stator
Fig. 11. 19
According to the forced vibration theory, a traveling wave B03 can be excited by four piezoelectric ceramic pieces. which are connected to voltage signals sinwt, coswt, - sinwt, and - coswt, respectively, and equally placed on the cylinder outer surface as shown in Fig. 11. 19. It is easy to change the phase difference between the signals when there is a need to change the direction of the traveling wave. After the above design was determined, dimensions of a stator could be devised. Since piezoelectric ceramic pieces are light enough to be neglected comparing with the cylinder in approximate solution, the primary size of the stator can be determined very quickly through the calculation of natural frequency by formulae in Chap. 1. Based on the primary size of the stator, a finite element model with 18000 elements. in which the stator uses Solid 15 and the piezoelectric ceramic pieces use Solid 5. was constructed with free to free boundary. After repeating the adjustment of the structure size. the final size of the stator with natural frequency 20. 8kHz of mode B03 was determined. as shown in Table 11. 1. The table also includes the property constants of material.
Ultrasonic Motors Technologies and Ap plicalions
336
Table 11. 1
Parts and material parameters of prototype radius/mm
Outer radius/mm
Length/mm
Density/ (kg/m' )
Duralumin
26
31
30
2 700
PZT-4
(Length)30
(Width)8
(Thickness) 1
7 600
Parts
Material
Stator PZT pieces
Inner
2. Design of other parts The size of cylinder rotor could also be determined after the determination of the stator. According to simulation, the closer the rotor lies to the stator, the higher driving torque will be. To validate the theoretical analysis and to take account of precision of machining, we made three rotors with different outer radii, as shown in Table 11. 2. Table 11. 2
Structure parameters of prototype
Rotor
Material
Inner radius/mm
"fa. 1
Duralumin
"fa. 2
Duralumin
"fa. 3
Duralumin
Outer radius/mm
Length/mm
23
25. 9
30
23
25.8
30
23
25. 7
30
Figure 11. 20 is the decomposed figure of the motor, in which the bushing and shaft were disposed for the output. Besides, to achieve the free-free boundary at two ends of the stator, an elastic cushion was installed at the bottom of the stator, as shown in Fig. 11. 21.
Fig. 11. 20 Decomposed figure of noncontact type USM with cylinder stator
11. 1. 4
Fig. 11. 21 Picture of non-contact type USM with cylinder stator
Performance Measurement of Non-contact USM
Since driving torque of the non-contact USM with cylinder stator is very low, we only measured the motor's speed using photoelectric method and the stall torque by combined method of high precision clectronic balance and a suspending weight. The other performance is untested because of limited testing means. The clockwise speed with rotor No.1 varies with frequency, as shown in
Chapter 11
Other Ultrasonic Motors
337
Fig. 11. 22. In the figure, the peak lies near to 18. 6kHz and the curve is steep in its left and right. Because of assembly and processing error, there is great difference between the clockwise speed with the maximum value of 2 100r/min and counterclockwise speed with the maximum value of 1 800r/min. 2500 2000 ~
.:;
"
~
1500
"::'
'"0.~ r/)
1000 500
q8.0 18.2
18.4 18.6 18.8 19.0 19.2 19.4 19.6 19.8 20.0 Freq uellcylkHz
Speed vs. frequency of noncontact type USM with cylindrical stator
Fig. 11. 22
11.1.5
Design of Non-contact Type USM with Disk Stator
Choosing B05 of the disk as operating mode with frequency of about 20kHz, the stator No.1 of a prototype was optimized and its results are shown in Table 11. 3. Table 11. 3
Parts and material parameters of prototype motor
Structure of mOlor
Material
Outer radius/mm
Inner
Thickness
Density/
radius/mm
/mm
(kg/m' )
Stator
Duralumin
15
a
1.0
2 700
PZT Pieces
PZT-4
45
30
0.5
7 600
Rotor
Duralumin
45
a
0.3
2 700
With a finite clement modcl meshed by 856 Solid 5 clements, as shown in Fig. 11. 23(a), the solution gained by A='JSYS finite clement software provided the nephogram of Bos mode with free boundary, in which there are five obvious wave crests as shown in Fig. 11. 23(b). Since prototype No.1 was validated whether it could operate, its practicality was not considered. After improvement, a practical prototype No.2 with planform (decomposed figures shown in Fig. 11. 21) was made. The stator was made of disk-shaped substrate and cylindrical piezoelectric ceramic element bonded to its bottom, and the rotor made of duralumin was also a disk. In order to stimulate the traveling wave of Bos mode, piezoelectric ceramic element was polarized in the same direction and was equally divided into twenty sec-
338
Ultrasonic Motors Technologies and Ap plicalions
(a) Fin ite elemenl model
Fig. 11. 23
(b) Mode nephogram
Finite analysis of disk stator
(a) Pl anfonll
Fig. 11.24
(b) Deco mposed figure
Non-contact type USM with disk stator
tor areas (sections). When twenty sections were connected to voltage signals wi th phase difference of 90 degrees, such as sinwt, cOswt, - sinwt, - cOswt, sinwt "', respectively from a four phase driver, the traveling wave of B05 mode will be excited. Here, w is elose to the natural frequency of B"mode. If the signals with phase difference of - 90 degrees were used as input, the traveling wave would travel in the opposite direction.
11.1.6
Testing of Non-Contact USM with Disk Stator
1. Mode testing of disk stator The frequency response curve shown in Fig. 11. 25 was obtained by PSV-300F-B, The operating frequency measured was 19. 6kHz, which was ncar 20. 8kHz obtained by calculation.
2. Driving testing When piezoelectric ceramic pieces were connected to voltage signal sinwt, coswt, -sinwt, and - coswt, with 300Vpp , From four-phase driver made by PDLab, the rotor would suspend and rotate elockwise at near the natural frequency 18. 2kHz and stop the rotating at near 20. 5kHz. When voltage signals were alterna ted with sinwt, - coswt, - sinwt, and coswt, the motor would rotate counterelockwise, which is the same direction as the traveling wave. Since levitation height of the rotor is so small that it cannot be seen by naked
Chapter 11
'"
Other Ultrasonic Motors
339
600
E
~ .,
."
.~ 0. E
<:
400 200
0
20
10
30
40
Frequcncy/kHz (a) Frequency response Curve
Fig. 11. 25
(b) Nephogram of mode Bos
Mode testing result of non-contact USM with disk stator
eye whether there is contact between the stator and rotor. According to thc opcrating mcchanism of both contact and non-contact type USMs, the following simple method can be used to check whether the rotor has a non-contact rotation: applying a pressure onto the rotor and then removing the pressure. If the rotation of the rotor becomes slow when a pressure is applied, and the rotation becomes fast when the pressure is removed. then the rotor has no contact with stator before applying pressure. The clockwise speed measured by photoelectric method varies with a frequency,as shown in Fig. 11. 26. in which the peak of the speed is near 18. 9kHz. On the left hand side of the peak, the curve is steep. Its maximum speed is 6 03lr/min and 5 986r/min in the other direction. Under the frequency of 18. 9kHz. stall torque measured is 3. 5 X 10- 5 :'\J. m which has the same order of the magnitude as simulation. 7000 6000 5000
.::
~
~
4000
~ ."
"8-
3000
en
2 000 1000 18.8 19.0 19.2 9.4 1 19.6 19.8 20.0 FrequencylkHz
Fig. 11. 26
Speed vs. frequency of non-contact type USM with disk stator
Listed in Table 11. 1 was performance data of the non-contact type USM in the world. From the table. it can be seen that the size of motor made by PDLab is
340
Ultrasonic Motors Technologies and Ap plicalions
rather big, but its speed is the highest. Table 11. 4
Compare with the present non-contact USMs of the same type
Research facilities
Dimension of Types of rotor stator/mm
Maximum speed /Cr/min)
Driving torque /C)I·m)
Yamagata University
20 XL 7
Paper blade
3 000
No report
Tokyo Institute of Technology
30 X 0.5
Metal disk
1 200
No report
NUi\i\
45 X 1. 5
Metal disk
6 031
3.5 X 10
11. 2
5
Linear Surface Acoustic Wave Motor
Research on surface acoustic wave (SAW) began in 1885 since English physical scientist Rayleigh published a paper with the title" On Wave Propagation along The Plane Surface of An Elastic Solid" based on his study on seismic wave-23 J • In the paper, the surface wave was illustrated in theory for the first time as a kind of wave traveling along the surface of half infinite elastomer where wave's energy concentrates. However, it had not been put into practical application owing to the limitation of scientific technology of that time before the 1960s. The development of planar semiconductor and laser technology and the availability of massive artificial piezoelectric crystals in the 1960s, laid a foundation for the development and application of the surface acoustic wave technology. From then, the surface acoustic wave technology has been gradually developed into an interdisciplinary subject which covers acoustics, optics, electronics, etc. Compared with the well known acoustic wave, transverse wave and longitudinal wave, the surface wave has its own features L21 -: its traveling speed is five orders less than electromagnetic wave, its wavelength is very short and the devices based on surface acoustic wave is easy to miniaturize; along its traveling path, signal can be stored and taken arbitrarily; surface wave devices can be easily integrated into a system. Since 1965 the interdigital transducers (IDT) were invented by American scientist White and Voltmov, surface wave technology has gained rapid development because of the useful features mentioned above. From the 1980s, SAW devices have been improved rapidly in America and especially in Japan. More SAW devices have appeared such as the filter, delay line, oscillator, acoustooptic modulator, touch screen, sensor, surface wave linear motor, etc.
11.2. 1
State of the Art
Surface acoustic wave USM is a type of micromotor. Unlike the USMs discussed in previous chapters which use bulk wave, surface acoustic wave USMs use acoustic wave traveling on solid surface. It has following features:" 28J: (1) Influenced very little by magnetic field and does not create any magnetic field. (2) It has a long drive span which can be more than 10cm.
Chapter 11
Other Ultrasonic Motors
341
(3) Its output force can reach more than 20:'\1 and has a speed more than 1. 5m/ s. (1) Its compact structure and small mass meet the needs of miniaturization. (5) It docs not nced lubricant, which is fit for applications in vacuum. (6) Likc USMs using bulk wave, it has quick responsc and a holding torquc whilc power is off; it can dircctly drivc a load without specd reduction gears and has excellent control characteristics. Surface acoustic wave USM has similar operating mechanism to the rotary type traveling wave USM. Compared with the rotary type traveling wave USM, it has own fcaturcs as follows: (1) Thc wa vc used for SAW USMs is surfacc acoustic wa vc. Whilc in rotary type traveling wave USM, traveling wave which is superposed by two orthogonal bcnding modcs of a beam is uscd. Sincc thc clliptical traces of points induccd by traveling wave are the samc along the dcpth direction of bcam, all points movc along the elliptical traces with the same amplitude and direction. In Rayleigh wave, the elliptical traces of points diminish exponentially with depth. Both types of motor can transmit energy through the contact between the stator and rotor(slidcr). Since the cnergy of SAW USM is concentratcd on thc surface of thc stator and that of traveling wave USM is scattcred through the stator, thc former has higher cfficicncy than thc the lattcr in thcory. Howcvcr, at prescnt efficiency of SAW USM is so low that it is less than 10 %. The detail reason will be discussed in operating mechanism of SAW USM. (2) Operating frequency (MHz) of SAW USM is far higher than the common USMs, which means that its opcrating wavelength is far shorter than the common USMs. So it is more convcnicnt for miniaturization. (3) SAW USM transducer has high energy density more than 100W/ cm', so this typc of motor can gcneratc high drive force and speed simultancously. (1) It is very convenient to fix a SAW USM stator, while it is usually very difficult for the common USMs discussed previously. (5) Temperature has little influence on synchronizing frequency of SAW USM transduccr. So thcre is no need for an auto frequcncy tracing system. Rcsearch on thc SAW USM began since thc 1980s. In 1989, for the first timc Sensor and Actuator Center (BSAC) of University of California at Berkeley made two prototypes of SAW USMs of linear type and rotary type, in which SAW was excited in pclliclc of ZnO and Si 3 :'\1, with thc frequcncy of 4MHz and slidcr was
made of spathic siliconL29 -. The fast flowing air layer between the stator and slidcr, caused by strong vibration of thc pcllicle, lcvitatcs thc slider above thc stator. Their driving forces arc very small bccausc thcy are not based on friction. In 1992, Kurosawa and his colleagues started to study SAW USM and made their first SAW USM with two dcgrccs of frecdom coo :. Its stator was madc of crystal Li:'\lb0 3 and slidcr was thrcc balls made of ruby, stcel and tungsten, rcspcctivcly. Thc planar motion of the slider was obtained by two orthogonal SAWs excitcd by IDT. In 1997, Kurosawa, Chiba, ct al. designed a multi-contact-point type lincar
Ultrasonic Motors Technologies and Ap plicalions
342
SAW USM L31- , as shown in Fig. 11. 27, which not only solves the pre-pressure problem between stator and slider but enlarges contact area.
(8)
AW U M
(b) Slider
M ulti-contact-point linear SAW USM and its slider
Fig. 11. 27
In 1997, Sano, Matsui, Shiokawa, et al. made experimental research[": on SAW USM using medium of liquid as shown in Fig. 11. 28. When SAW was excited in stator by IDT, the steel ball slider will move along the direction of traveling wave. Besides Japanese researchers. Helin and his colleagues in France Valenciennes University did the research on dynamic mathematic model of SAW USM. Woods[oo: in England Sheffield University. Breedveld in Holand Twente University and his colleagues also had done a few researches on SAW USMs:3<] (see Fig. 11. 29). Steel ball slider lOT
rv
SAW
DODD Stator
_
" { ) /water
SAW USM using medium of water
Fig. 11. 28
I .
Fig. 11. 29 Linear SAW USM designed by TWENTE
The actuator used in spaceships and satellites requires not only a high torque and quick response but also small volume, light in mass, and long life span. Therefore. it is the important aim of NASA for a long time to provide spaceship with miniature actuator with good performance. Fig. 11. 30 shows the sketch of SA W USM with multi- DOFs made by JPL of America:"]. In China. :'\Janjing University and Zhejiang University have researched on the SA W USM.
In 2002. Guangming Zhang of acoustic laboratory in Nanjing
U ni versi ty made a rotary type SAW USM[36: as shown in Fig. 11. 31. Its drive frequency is 9. 85MHz and the stator was made of Li:'\JbO, . Two groups of IDT
Chapter 11
Other Ultrasonic Motors
343
Screw
x
Stator
Fig. 11. 30
lOTI
IDT4
Fig. 11. 31
Sketch of multi-dimensional SAW USM
Rotor
Small steel ball
Shaft
IDT2
IDT3
Rotary SAW USM designed by Guangming Zhang
on substratum excite two SAWs parallel to each other with opposite direction to drive slider and has the maximum velocity of 180r/min.
11.2.2
Surface Acoustic Wave and Its Generation
SA W has many typcs: 37 ] according to different boundary conditions, such as Rayleigh wave, electric acoustic wave, Rayleigh wave, and lamb wave. In the SAW technology, Rayleigh wave which travels along the surface of elastomer has the widest applications. Unlike transverse wave and longitudinal wave, it is generated by superposition of transverse and longitudinal waves under the boundary condition of zero stress surface. Compressed or expanded by longitudinal wave and sheared by transverse wave simultaneously, points on the surface move along elliptical trajectory and have the maximum amplitude which diminishes exponentially with depth. Therefore, 90 % of its energy is concentrated in the depth of one wavelength from the surface, and in some range the direction of elliptical motion changes, as shown in Fig. 11. 32. There are many ways to gain SAW. One simple and efficient way is to use IDT. Here the operating mechanism of IDT is interpreted by the example of piezoelectric crystal Li:'\lbO, substratum. The converse piezoelectric effect in Li:'\lbO, substratum can be written as
344
Ultrasonic Motors Technologies and Ap plicalions
51 52 53 54 55 56
0
- d 22
0
d 22
0
0
d 31 d 31 d 33
0
d 15
0
d 15 - 2d 22
0
0
0
0
r~,
(ll.5)
E3
where there are only two eleetric field parameters. as shown in Fig. 11. 33. According to the above formula. strains on the surface of LiNb0 3 substratum caused by the electric field of IDT, are 51 , 52' 53' and 54. 53 generated by E3 corresponds to acoustic longitudinal wave. which has a displacement wand travels along the z axis. 54 generated by E2 corresponds to acoustic transverse wave, which has the displacement of v and travels along the y axis. Therefore. when boundary condition is free on the surface , SAW will be excited by IDT in the surface layer of yz shear type LiNb0 3 substratum. Rayleigh wave prop.gation di rect ion
Fig. 11. 32
Point motion
Motion trajectories of points in the elastomer
+
+
+
Fig. 11. 33 Electric field distribution in the yz shear type LiNb0 3 using IDT for excitation
IDT is used for transducer in linear SAW USMs discussed in this book, which are two-port devices with interlaced and connected electrodes- 37 - 39J , as shown in Fig. 11. 31. Its main parameters inelude the pairs of fingers N, hole radius W. cyele length L. finger width a and finger gap b which is equal to a. Since interdigital electrodes are arranged periodically in both polarity and position, an elec-
Chapter 11
Other Ultrasonic Motors
345
tric field will be set up in the substratum when the electrode is connected to alternating voltage. Because of anti-piezoelectric effect. each pair of interdigital electrodes generates SAW in substratum and the SAW excited by IDT is the superposition of SAWs generated by all pairs of interdigital electrodes.
Cross seclion
(I
b
Ichnography
Fig. 11. 34
Sketch of IDT
If the SAWs excited by each pair of interdigital electrodes have the same amplitude and there is no decay during traveling. since electrodes are arranged periodically. the phase difference of SAW excited by two adjacent pairs of electrodes is
M= wL Zv,
(1l.6)
where v, is the propagation speed of the SAW and w is an exciting angular frequency. The intensity of SAW excited by IDT with n intcrdigital electrodes can be written as (ll.?)
where the positive and negative signs of two adjacent items in square bracket are caused by opposite polarity of adjacent fingers. and Eo is amplitude of SAW excited by each pair of interdigital electrodes. At 6.{) = wL/(ZvJ = rr. total output E, reaches maximum (ll. 8)
In this case. the acoustic synchro angular frequency of IDT is Wo cycle length is L
=
Zrrv, WO
=
v.' fo
=
AD
=
Zrrvj Land (1l.9)
where AD is the wavelength of SAW. From the above discussion. it is concluded that the intensity of SAW excited by IDT depends on pairs of fingers. The more pairs there are. the stronger intensity of the SAW is. And the acoustic synchro angular frequency depends on finger width and finger gap. When frequency of exciting voltage is equal to acoustic synchroangular frequency WO' SAW is the strongest. When exciting frequency is not equal to acoustic synchro angular frequency. that is. w # WO' using w = Wo + 6.w as well as Eq. (ll. 6) and Eq. (ll. 7). total output is simplified as E,
ZNE 0 smx ei(W'-EJ x
(11.10)
346
Ultrasonic Motors Technologies and Ap plicalions
where.1': = Nrrt::.w/wo and N = n/2. From this. it is concluded that IDT has fundamental performance as follows L38 . : (1) The output of IDT is the function of frequency. varying with sinx/ x regularly. as shown by the frequency response of equal finger IDT in Fig. 11. 35. The distance between the first pair of zero points is 2t::.w/ wo = 2/ N. It shows that the bigger number of period is or the more pairs of fingers are. the more narrow frequency response width is.
Relative amplitude
-21N -liN
Fig. 11.35
o
liN
21N
/',.wlwo
Frequency response of IDT relative amplitude
(2) Even using the acoustic synchro angular frequency to excite the substruction. the amplitude of the substratum is so tiny (of the order of nano meter). Therefore the speed of SAW USM cannot be controlled by adjusting frequency. (3) The intensity of SAW excited by IDT is proportional to the pairs of interdigital electrodes N. The greater N is. the stronger SAW is. At the same time. the longer the overlap length of fingers is. the stronger SAW is. (1) The frequency characteristic of IDT is closely related to the geometry of IDT. The cycle length determines the operating frequency of IDT. (5) IDT is the linear device operating with small signal and satisfies the reciprocal principle. that is. the transmitting and receiving characteristics are the same. (6) The phase of SAW excited by IDT varies linearly with frequency. Besides. it is easy to fabricate IDT by semiconductor production process. which has a high machining precision. This makes mass production convenient.
11.2.3
Operating Mechanism
The operating mechanism of SAW USM is that the SAW excited by IDT causes the elliptical motion of points on stator's surface. and this elliptical motion pushes the slider through the liquid or friction between the stator and slider. Its structure includes three parts: a piezoelectric substratum. two IDTs. and a slider. as shown in Fig. 11. 36 L10 .• The Rayleigh wave is excited by IDT and propagate on the surface of the substrate. The surface points on the substrate move in elliptical motion as illustrated in Fig. 11. 37[<0:. When a slider is placed on the stator under suitable pre-pressure. the friction force between the stator and slider pushes the slider to move in the opposite direction to the traveling wave.
Chapter 11
Other Ultrasonic Motors
347
Powc r sourcc IDT Stator Pre-pressure Slider IDT
Elliptical motion of points on the surface of the stator
Fig. 11. 37
Fig. 11. 36 Sketch of linear SAW USM
1. Stator The performance of SAW devices depends mainly on the characteristics of piezoelectric material, which is required to have high electromechanic coupling and low encrgy wastc whcn wavc travels along it. Besides, of course, the piezoelcctric matcrial used in stator should bc rcpcatable, reliablc and even in property, cheap in pricc, and casy to producc. Three common types of piezoelectric materials used for SAW devices are piezoelectric crystal, piezoelectric ceramic and piezoelectric thin film.
2. Slider The material and structure of the slider have much influence on the output characteristics of SAW USM. Since thc linear SAW USM is drivcn by frictional force, the output forcc of the motor increases with the incrcase of contact arca betwccn thc slider and thc stator with a fixcd pre-prcssure. On the surfacc of thc substratum, the surface roughness and vibration amplitude of surface points have the same order of magnitude (nanometer). Also, the contact areas are influenced by compressed air layer, oil, dust, etc. Therefore, the pressure between stator and slidcr must bc increased. Contact point
~rl'________6_m_m______~ 6 mm
4.2 mm
Fig. 11. 38
Contact slider using ruby
Stee l ball with diameter 0.201111
Fig. 11. 39
Multi-contact-point slider
In 1994, Kurosawa and his collcagucs in Tokyo Univcrsity madc a lincar SAW USM whosc slider is the small ball madc of stcel, hard alloy or ruby as shown in Fig. 11. 38- 30J • The high pressure up to IGPa and small contact area between stator and slidcr madc thc output forcc rathcr small (about 1mN). To increasc thc contact arca, in 1997, Kurosawa and his collcagucs used a slider with multi-contact-points as shown in Fig. 11. 39- ' 1J . It consisted of an aluminum plate with the dimcnsion of 6mmX 6mm and 500 stcel balls with diametcr of o. 2mm sticking on
Ultrasonic Motors Technologies and Ap plicalions
348
the plate. The contact area is 2. 5 X 10- 3 mm 2. Under the same pressure as the single ball slider, its output force increased greatly. To further increase the contact area between the stator and the slider and to improve performance of the motor, in 2001, Kurosawa developed a slider using silicon, as shown in Fig. 11. 27 (b), which had many projections with different diameters made by silicon surface processingL12-13-. The output force and speed of the motor using this type of slider increased 12 times compared with the multicontact-point type. Table 11. 5 shows the comparison of six silicon sliders with multi-contact-point slider. It can be seen that the contact area of silicon slider is 50-1000 times greater than multi-contact-point slider. Table 11. 5
Comparison between silicon sliders and multi-contact-point slider
Type of slider
Diameter of proj eetions/ I-'m
Density of projections
Contact area of
/CP/mm2)
slider/mm 2
Type 1
50.00
90
2. 84
Type 2
30.00
90
1. 02
Type 3
20.00
90
0.45 0.11
Type 4
10.00
90
Type 5
20.00
380
1. 91
Type 6
20.00
20
O. 10
Multi-contact-point slider
2.52
14
O. 002 5
3. IDT IDT is the key part of SAW USM and is one of the most efficient methods to excite SAW. It determines the amplitude of Rayleigh wave on the stator surface and is important for improving the operating efficiency and decreasing the input power. The efficiency of the motor is comparatively low when the IDT shown in Fig. 11. 31 is used. The reason is that when a high frequency voltage is applied to IDT, the SAW is generated and travels towards both sides. The wave needed for driving the slider is disturbed by the wave reflected by the edge. Thus, the efficiency of the motor decreases a lot (usually about 1 %). The SAW motor using an energy circulation method shown in Fig. 11. 10 was used to improve the efficiency of IDT:"]. Fig. 11. 40 (a) shows its structure, which consists of two driving IDTs and two one-way IDTs (also known as UDT) connected to each other. Rayleigh wave is excited by two driving IDTs on stator surface and when it reaches the other end, one group of UDT absorbs it and converts it into electric energy, which is used for the other group of UDTs to generate Rayleigh wave. Power synthesizer type energy circulation structure, as shown in Fig. 11. 10 (b), consists of one power synthesizer and two groups of UDTs. Two groups of UDTs are connected to the input and output ends of power synthesizer, respectively. The power absorbed by one group of UDT can be transmitted to the other group of UDT by the power synthesizer. Both methods can be used for energy circulation to increase the efficiency of motor. The linear SAW USM using the method as shown in Fig. 11. 40(a) can reach the efficiency of 5%[,,<5:.
Chapter 11
Tran ducer Eocos (&f) Eosin ((
Other Ultrasonic Motors
UDT
Rcnecl ing ~
(a)
Fig. 11. 40
349
Slalor
machi ne
(b)
Stator using energy circulation method
References [ 1
J
[ 2
J
[ 3
J
[ 4
J
[ 5
J
[ 6
J
[ 7
J
[ 8
J
[ 9
J
[10J
[llJ [I2J
[13J
[l1J [15J
[I6J [17J
[I8J [19J
Chunsheng Zhao. Ultrasonic motor techniques [or 21 st century. Engineering Science, 2002, 1(2): 86-91. (in Chinese) Cunyuc Lu, Chunshcng Zhao. :'-lew development of piezoelectric motor with coupling mechanism. China Mechanical Engineering, 2003, 11(7): 626-629. (in Chinese) T Morita, T Niino, H Asama. Rotational fccdthrough using ultrasonic motor for high vacuum condition. Vacuurn, 2002, 65: 85-90. K Asai, M K kurosawa, T Higuchi. Surface acoustic wave motor usin energy circulation driving mcthod. IEEE Proceedings of Ultrasonics Symposium, 2001,1: 525-529. K :'-Iakamura, T Ito, M Kurosawa. A trial construction of an ultrasonic motor with fluid coupling. Jpn. J. Appl. Phys., 1990, 29(1): 160-161. S Hirosc, Y Yamayoshi, H Ono. A small non-contact ultrasonic motor. IEEE Proceedings of Ultrasonics Symposium, 1993,1: 153-156. Y Yamayoshi, S Hirose. Ultrasonic motor not using mechanical friction force. International Journal of Applied Electromagnetics in Materials, 1992, 3: 179-182. H Hanada, H Iida, K Nakamura. Radiation piezoelectric driving ultrasonic motor for wing of end fccdcr. Winter in Heisei 3, Nion Proceeding, 1991, 2-1-3: 937-938. J unhui H u, :'-Iakamura K. Characteristics o[ a noncontact ultrasonic motor using acoustic levitation. IEEE Ultrasonic Symposium, 1996,1: 373-376. Junhui Hu. Research on Non-contact Driving High Speed Ultrasonic Motor. Dissertation [or the Degree o[ Doctor o[ Philosophy. Tokyo: Tokyo Institute o[ Technology, 1997. Jingquan Liu, Boda Wu, Zhigang Yang, ct al. A new type o[ circular cylindrical non-contact ultrasonic motor. Acta Acustica , 2001, 3(2): 113-116. (in Chincsc) Changliang Xia, Junhui Hu, Tingna Shi, ct al. Study on theory and experiment of non-contact type ultrasonic motor with fluid medium. Proceedings of the CSEE, 2001, 21 (8): 6467. (in Chinese) B Chu, R E Ap[el. Acoustic radiation pressure produced by a beam o[ sound. Journal of the Acoustical Society of America, 1982,72(6): 1673-1987. S U cha, Y Hashimoto, Y Koikc. :'-Ion-contact transportation using ncar-field acoustic levitation. Ultrasonics, 2000,38: 26-32. W L Nyborg. Acoustic streaming near a boundary. Journal of the Acoustical Society of America, 1958, 30(4):459-467. C P Lcc, T G Wang. Ncar-boundary scrcaming around a small sphcrc due to two orthogonal streaming waves. Journal of the Acoustical Society of America, 1989, 85(3): 1081-1088. Ye Ji. Research on Non-contact Type Ultrasonic Motor. Dissertation [or the Degree o[ Doctor o[ Philosophy. Nanjing: :'-Ianjing University of Acronautics and Astronautics, 2006. (in Chincsc) Chunshcng Zhao, Yc Ji. Cylindcr typc non-contact ultrasonic motor. Chinese Invention Patent, CN2004100701. X, 2004-11. (in Chinese) Chunsheng Zhao, Ye Ji. Disk type non-contact ultrasonic motor. Chinese Invention Patent, CN2001100702. 1, 2001-11. (in Chinese)
350 [20J [21J [22J [23J [24J [25J [26J [27J [28J [29J [30J
[31J [32J [33J [31J
[35J [36J [37J [38J [39J [10J [41J [ 12J
[43J [11J [45J
Ultrasonic Motors Technologies and Ap plicalions Ye Ji, Chunsheng Zhao. Modal shape measurement of cylinder stator in non-contact ultrasonic motor. Journal of Vibration, Measurement & Diagnosis, 2005,25 (1): 1-3. Y c Ji, Chunsheng Zhao. Cylinder type non-contact ultrasonic motor. Journal of Nanjing University of Aeronautics & Astronautics, 2005, 37(6): 690-693. (in Chinese) Ye Ji,Chunsheng Zhao. A new type non-contact ultrasonic motor with high revolution speed. Piezoelectrics & Acoustooptics, 2006 28(5): 527-529. (in Chinese) L Rayleigh. On wave propagation along the plane surface of an clastic solid. Proc. London. Math. Soc. , 1985, 17: 4-1l. Changkun Xu. Surface Acoustic Wave Apparatus and Its Application. Beijing: Science Press, 1981. (in Chinese) M K Kurosawa. State-of-the-art surface acoustic wave linear motor and its future applications. Ultrasonics, 2000, 38(3): 15-19. M Takasaki, M K Kurosawa, T Higuchi. Optimum contact conditions for miniaturized surface acoustic wave linear motor. Ultrasonics, 2000, 38(3): 51-53. P Helina, V Sadaune, C Druon. Theoretical and experimental study of linear motors using surface acoustic wave. Sensors and Actuators A: Physics, 1998, 70 (0): 67-71. T Shigcmatsu, M K Kurosawa, K Asai. :'-Ianomctcr stepping drives of surface acoustic wave motor. Ultrasonics, 2003, 50(4): 376-385. R M Moroney, R M White, R T Howe. Ultrasonic mieromotor. IEEE Ultrasonics Symposium, 1989,2: 745-748. M K Kurosawa, M Takahashi, T Higuchi. Ultrasonic linear motor using surface acoustic waves. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 1996, 13 (5): 901-906. M K Kurosawa, M Chiba, T Higuchi. Evaluation of a surface acoustic wave motor with a multi-con tact-point slider. Smart Mater. Struct. , 1998,7: 305-31l. A Sano, Y Matsui, S Shiokawa. A new manipulator based on surface acoustic wave streaming. IEEE Proceedings of Ultrasonics Symposium, 1997,1: 467-470. http://www. sheL ac. uk/eee/resreport/full/semi/semil9. htm1,2006-03-2l. P J Feenstra, PC Brccdvcld. Modelling and experimental validation of a linear surface acoustic wave motor prototype. 8'" Mechatronics Forum International Conference. Netherlands: Drebbel Institute for Meehatronies, University of Twente, 2002: 231-240. Improved multiple-DOF SAW piezoelectric motors. http://www.nasateeh.com/briefs/ fcb03/:'-IP020859. html L P Cheng, G M Zhang, S Y Zhang, et al. Miniaturization of surface acoustic waves rotary motor. Ultrasonics, 2002, 39:591-594. Ming Chen, Dongyuan Fan, Suilao Li. Surface Acoustic Wave Sensors. Xi'an: Northwestern Poly technical University Press, 1997: 1-5. (in Chinese) Mingshan Xiao, Daoren Song. Basement of Surface Acoustic Wave Apparatus. Jinan: Shangdong Science Technology Press, 1980: 56-59. (in Chinese) Guidong Luan, Jinduo Zhang, Renqian Wang. Piezoelectric Transducer and Transducerarray. Beijing, Peking University Press, 1990. (in Chinese) M K Kurosawa, H Itona, K Asai. Elastic friction drive of surface acoustic wave motor. Ultrasonic, 2003, 41(6): 271-275. M K Kurosawa, M Chiba, T Higuchi. Evaluation of a surface acoustic wave motor with a multi-contact-point slider. Smart Mater. Struct, 1998, 7: 305-31l. :'-I Osakabc, M K Kurosawa, T Higuchi, et al. Surface acoustic wave linear motor using silicon slider. Proceedings of IEEE Workshop on Micro Electro Mechanical Systems. Heidelberg, 1998: 25-29. M K Kurosawa, :'-I Osakabe, K Tojo, et al. Sur face Acoustic Wave Linear Motor with a Silicon Slider. Technical Report of IEICE, 1998: 55-62. K Asai, M K Kurosawa. Surface acoustic wave motor using an energy circulation driving method. IEEE Ultrasonic Symposium, 2001, 1: 525-529. K Asai, M K Kurosawa, T Higuchi. :'-Iovel power circulation methods for a surface acoustic wave motor. IEEE Ultrasonics Symposium, 1999, 1: 667-670.
Chapter 12
Driving Techniques for Ultrasonic Motors A disk-type traveling wave rotary ultrasonic motor is a kind of USMs that has bccn widely used in many areas becausc its drive and control techniqucs arc relatively mature[l]. Provided no exception, this and the following chapter will take the traveling wave USMs as an example to expound the drive and control techniques, which will be refercnccs for othcr typc of USMs.
12. 1
Design Requirements for Drivers
As we know, driving USM needs to apply power with ultrasonic frequency. A driver is important for application of USMs. The quality of the driver affects the motor's output performance and its applications. Since USM is a capacitive load for the driver, some design requirements are listed here L2 -6J : (1) The driving frequency of USM depends on the resonance frequency of the stator of motor. For various stators with differcnt structurc and sizc, the rcsonance frequency is generally within 20-100kHz, so the output frequency of the driver must accord with it. (2) USM requires that its driver must provide 2 (or 1) phase voltages with ultrasonic frcquency and 90° phase diffcrcnce. Thc output voltages of thc drivcr must be the same frcquency and amplitudc, and the peak-to-pcak amplitude is generally tens to hundreds volts. The driver actually can be regarded as DC-AC inverter. (3) USM is a capacitive load of the driver. To guarantee the motor to operate efficiently and safety, a matching circuit must be addcd between thc driver and the motor. The matching circuit can make the driver transfer efficiently energy to thc motor; on thc othcr hand, it can also improve the waveform of the output voltage to reduce inter fer entia 1 harmonics. Therefore, the voltage applied to the motor is nearly a pure harmonic wave, and to avoid interference modes of the stator. (4) Thc operating frequency of USM should be selcctcd ncar thc anti-rcsonancc frequcncy of the stator's admittance frcquency characteristic. Of coursc, if the operating frequency is near the mechanical resonance frequency, a low voltage could drive the motor cfficicntly sincc the impedance of the stator is minimal at this point. But the authors found that the motor's comprehensive performances are better when it operates near the anti-resonance frequency_3-. A higher voltage can incrcase thc output powcr of thc motor. However, tcmperature in-
C. Zhao, Ultrasonic Motors © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011
352
Ultrasonic Motors Technologies and Ap plicalions
creases seriously in consequence. This will have a strong impact on the motor's efficiency, and thc motor will opcrate unstably. (5) USM's speed can be adjusted by a control loop. To meet this requirement, we can adjust the frcquency and amplitude of output voltage, or thc phasc diffcrcncc bctween the 2 (or 4) phasc voltages (scc Chap. 13). All thcse parameters must be continuously tunablc. In addition, thc limit circuits for the frcquency, voltage, and current, are necessary to ensure the motor works normally. The driving techniques of USMs are being developed with the improvement of thc ultrasonic motor thcory. Expandcd applications bring forward highcr demands for USMs, shown as followsL7-17J :
1. The miniaturization of the driver Currently, the volume of a driver/controller is relatively big compared with the USM, which restricts its promotion. Therefore, the miniaturization of the driver/ controller is the demand for expanding its applications. Thc currcnt methods to reduce the volumc of the driver includc: (1) Changing thc way of using discrctc componcnts to build thc drivcr. Many parts of the driving circuit, such as the signal generator, frequency divider and phasc splittcr, and control circuit can bc integrated into FPGA/CPLD, MCU, DSP, or ASIC, this could make the driver/controller compact. (2) Investigating ncw driving/controlling techniqucs. Discarding bulk transformers and matching circuit, integrating the driver/controller and USM could make the system smaller. At present, the development of micro systcm, such as MEMS, micro-robot, and micro-satellite, provide excellent opportunities for the miniaturization of the USM and driver/ controllcr.
2. Novel driving techniques The author considers that the study on the new type driving technique of the ultrasonic motor should be developed from three aspects as follows: (Dusing piezoelectric transformers to replace thc high frcquency transformcr and matching circuit can make the motor and the piezoelectric transformer to integrate in a design[12: ; CZ)study on ncw sclf-cxcitation oscillating drivcrs[8] , which can sclf-adaptively corrcct driving signals according to thc varicty of ultrasonic motor characteristics and naturally realize the automatic frequency tracking function, and simplify driving control circuit; ® study on thc driving techniquc with ncw driving mechanism oriented to the micro ultrasonic motor. The existing driving mechanism cannot fit the technique request, so the study on driving technique for micro electromechanical system becomes more and more important and urgent. Figurc 12. 1 is the basic diagram of USM driver, which is mostly used at prcscnt. It is mainly composed by three parts: a signal gcncrator, frcquency divider and phase splitter, and power amplifier/matching circuit- 2-. The signal generator creates a base square-wave signal, whose frequency is variable. The frequency dividcr and phasc splittcr circuit producc 4 channel square wavcs with 90 phasc difference each other. The power amplifier/matching circuit amplifies aforementioned square waves and forms a sinwt and cOSwt with high voltages to drivc 0
Chapter 12
Driving Techniques for Ultrasonic Motors
353
USM. The following sections will describe the three mentioned parts in detail.
Fig. 12. 1
12.2
Block diagram of the driver of ultrasonic motors
Signal Generator
As mentioned above. the function of a signal generator is to create a base squarewave signal whose frequency is variable. The operating principles of three commonly used generators arc introduced as follows.
12. 2. 1
RC Multivibratorlls-19J
Figure 12. 2 (a) is a square-wave generator composed by three inverters (G1 ( 3 ) , two resistors. and a capacitor. Rand C make up delay circuit. R, is a current-limiting resistor with low resistance around lOOn. Fig. 12. 2(b) is the waveforms of the RC multi vibrator. The oscillating period of the oscillator is T= 2. 2RC. and the frequency f = 1/2. 2RC. Any change of R or C will alter the frequency of the oscillator. Since the USM can be considered as a capacitor. it can replace the C of the RC oscillator to achieve resonance. Choosing suitable value of R, the frequency of the os-
u,
[/1
"'!LJLJ
•t
o
"~!JLJL •I
o U3
of---tf-------tf'-------Ca) Circuit diagram
Cb) Waveforms
Fig. 12. 2
RC Multivibrator
354
Ultrasonic Motors Technologies and Ap plicalions
cillator can be set to equal to the operating frequency of the motor, and the motor can run at this frequency. But this way is only suitable to drive a single-phase USM with low power.
12.2.2
555 Muitivibrator-1819J
Figure 1Z. 3 is an oscillator based on 555 timer Ie. R 1
,
R"
and C are external
components. When the pin 5 of the timer is floated, the resonance frequency is
f
=
_1_ t1 +t,
=
1.13 (R 1 +ZR,)C
(1Z. 1)
The voltage on pin 5 is the threshold voltage of U 1 and U,. Alteration of this voltage can also change the resonance frequency of the oscillator.
555
. . . . . ... ...... ... . ..:
7J3~~ b:tm 1/3U
~
~ l·············r····· l-··········
l:
~
~
;1
ce ·· __ ····--···T······
o v.
v.
[ ~
~ ~
~---;'
~ ~ l
.
~ j
~
1
r----;
O ~--~---+-----L--~----~
- 1,--
(a) Circliit diagram
I,
r-
(b) Wavefonns
Fig. 12.3
555 Multivibrator
The voltage given to pin 5 is kU" (O
, t1
=
k-Z (R 1 +R,)ln Z(k-1)
(1Z. Z)
And the time for decreasing U, from kU" to kU,,/Z is t, """ R,C InZ
=
O. 7R,C
(1Z. 3)
Therefore the output frequency is
f= t:+1 t,
(1Z. 1)
Figure 1Z. 4 shows the relationship between the output frequency and the voltage on pin 5 of the 555 timer. It is shown that they have good monotonic relationship, which is very suitable for stabilizing the motor's state using isolated electrode feedback.
Chapter 12
Driving Techniques for Ultrasonic Motors
355
150,-----,-----,----,,----,-----,
100
50
o
k
Fig. 12. 4
12. 2. 3
Output frequency vs. control voltage
Voltage Controlled Oscillatoi 5J
The 3rd method to generate a base signal is to use a voltage controlled oscillator (VCO) , whose frequency is controlled by the input voltage. Usually LM331 or CD4046 is used for VCO, the upper frequency of the former device is 100kHz, and that of the latter is 1. 2MHz. Now, eD'1016 will be taken as an example to explain the method of using veo.
f
U.. 9 CD4046
c
fn1A~
---- -- --- ---- -- --- ----- ----- -- ---- -
6
7 .-------¢ I I 12
4
U• frrrln
II, o ~------~------~----
V,,12
(a) Circuit diagram
U..
(b) Output frequency v .input vohagc
Fig. 12. 5
eD4046 veo
The CD4046 is aetually a phase-locked loop (PLL), it contains a RC type veo. An external capacitor C and resistor RJ are used for charging and discharging elements of the veo, by which the generated frequency is proportional to the control voltage (U in ) from pin 9. When the U in equals to 0, the output frequency is minimum value (fmin), and when the U in equals to the supply voltage Vee' the output frequency increases linearly to the maximum one (fmox) . R j , R, , and C determine the range of the VCO oscillation frequeney. The CD4046 VCO has the characteristics of a simple configuration, wide range of output, good linearity, and low temperature coeffieient, whieh make it to beeome a widely used signal generator in the USM drivers. However, with the improvement of performanee and cost reduction of MCU and DSP, using them for signal generator will be a trend.
356
12.3
Ultrasonic Motors Technologies and Ap plicalions
Frequency Divider and Phase Splitter
USMs need two phase orthogonal signals. The phase splitter can convert a single phase signal to two phase signals. Generally. the phase splitter is always combined with frequency divider. So this part of circuits is called as a frequency divider and phase splitter (FDPS). The principles of two FDPSs arc introduced as follows.
12. 3. 1
FDPS Composed by Shift Register[<]
The present power amplifier of the driver uses push-pull or bridge circuit. each driving signal has the phase difference of 90° each other, as shown in Fig. 12. 6. The sequential relationship of the signals is 1001
I
-
1100
-
0110 -
0011 I
Apparently. one cycle is the process of cyclically shifting the numbers toward right. This function can be implemented by a shift register. Shifting the numbers toward left will change the rotation direction of the motor. Therefore, a 1-bit bidirectional universal shift register can implement a frequency divider and a phase split functions. its schematic diagram is shown in Fig. 12.7.
A+h
n n on n rA-) n n 0
. I
H+)
•I
v"
a
CWfCCW(
Base signal
Fig. 12. 6
Waveforms of FDPS
CD40194
D, D2
Qo Q, Q,
D,
Q;
So S,
SOR SOL
Do
Fig. 12.7
A+ 8+
8-
CP
FDPS composed by CD40194
In Fig. 12. 7, the base signal from signal generator is inputted to CP port of CD10191. Do-D3 are numerical inputs. SDR and SDL are the inputs for rightshift and left-shift. Connecting them with corresponding outputs can fulfill cyclically shifting. Ports So and S1 can control the shifting direction, and the rotation direction of the USM sequentially. We know from the function of this circuit that
Chapter 12
Driving Techniques for Ultrasonic Motors
357
when the input signal is converted to the signals shown in Fig. 12. 6, its frequency is dropped to a quarter. Therefore. the frequency of a base signal must be 4 times of the operating frequency. Using double D flip-flop and exclusive-OR gate (shown in Fig. 12. 8(a)) or Up/Down Counter (shown in Fig. 12. 8(b)) can also construct FDPSs, their detailed operating principles arc omitted here.
Base signal
Base signal
CW/CCW Con1rol
CP
UID
CW/CC W Cowol
Fig. 12. 8
Other FDPSs
The duty cycle of every driving signal in Fig. 12. 6 is 50 %. If these signals arc used to directly drive switching elements in push-pull or bridge circuit, direct pass failure may occur since the switching characteristics of the switching elements are not the same and they may be damaged. It is very dangerous for the driver. Therefore, a dead-zone must be added to the driving signal shown in Fig. 12. 6 to make its duty cycle less than 50 %. There are two methods to set a dead-zone L2 - : (1) Integral method Figure 12. 9(a) is the schematic diagram of setting the dead-zone using integral method and 12. 9 (b) shows the waveforms on every node. When the driving signal A passes through an integral circuit composed by Rl and Cl in Fig. 12. 9 (a) , the rising edge of the A turns into the waveform of B. After compared with a fixed voltage, it becomes the waveform D. Moreover, when the falling edge of A comes, the transistor P I breaks over and the capacitor C l discharges immediately. This process will delay the rising edge of A, maintain the falling edge and then form the dead-zone. The range of the dead-zone can be adjusted by the voltage of node C. (2) D flip-flop method Figure 12. 10(a) is the schematic diagram of the dead-zone set up by D flipflop method, and 12. 10(b) shows the waveforms on every node. The rising edge of A sets the D flip-flop, and then charges the capacitor C l • When the voltage of node D rises to the threshold of the device, the D flip-flop is reset. and output low voltage. Since node C is the "NOT" of node B, and node E is the" AND" of node C and A, the dead-zone is then formed. The range of the dead-zone can be adjusted by resistor Rl and capacitor C 1 •
Ultrasonic Motors Technologies and Ap plicalions
358
Ca) Schematic diagram
Fig. 12. 9
Cb) Waveforms of nodes
Dead-zone circuit set up by integral method
:1 v.. D SET Q ....:A~
B
_ _p---I>
Base signal
:1 n Ch 0 0
:1 / I £1 0 (a) Schematic diagram
Fig. 12. 10
12. 3. 2
r-I I
•I •I I
I
•I
(b) Waveforms ofnodes
Dead-zone circuit set up by D flip-flop method
FDPS Composed by CPLD 20 21
The above circuits can only generate driving signals with the dead-zone for fixedtime, i. e. the range of the dead-zone is irrelative with the signal frequency. In applications however, using driving signals with 40 % duty cycle for all frequencies to drive the USM can achieve higher efficiency. Utilizing the aforementioned discrete devices is difficult to fulfill this object, but a complex programmable logic device (CPLD) is competent for this task. CPLD has the advantages of high performance, high density of integration,
Chapter 12
and easy development.
359
Driving Techniques for Ultrasonic Motors
It
IS
designed and simulated with software MAX
+
PLUSII of ALTER A Co .. To develop the CPLD, graphies and programming language are used. Fig. 12. 11 is the simulation results of CPLD. In this figure, elk is a elock whose frequency is adjustable, cw and ccw are control signals for CW, CCW, and STOP (STOP when CW and CCW are inphase; CW or CCW when they are out of phase), and qo - q3 are driving signals with 10 % duty cyele to drive the power transistor. Fig. 12. 12 is the actual waveforms of driving signal and output voltage. 20.0 fls
D-- clk
0
D-- cw
0
40.0 JJS
60.0 fls
80.0 fls
100.OflS
120.0flS
D-- ccw
-aq3
0
-aq2 -aql
0
-aqo
0
Fig. 12. 11
Simulation waveforms of CPLD
e> >
> (5 ~
~
-
N
G
:;:
u
51-lsIDiv
Fig. 12. 12
12.4
Waveforms of driving signal and output voltage
Power Amplifier Techniques
Ultrasonic motors need the signals with a high freguency, high voltage, and certain power. Small signals generated from the oscillator have to be amplified. This means that the generated unipolar square wave should be transformed into bipolar square wave to drive high-frequency transformers. The power amplifier components are mainly power transistor, MOSFET, and 1GBT. Since USM usually has low power, and operates at relatively high frequency, MOSFET is suitable for such application. The MOSFET device has a high input impedance of up to 40MD. It is a volt-
360
Ultrasonic Motors Technologies and Ap plicalions
age-drive type switching device. When the grid-source voltage is greater than the threshold, the MOSFET is on. Otherwise it is off. It is relative simple to drive MOSFET. In some occasions, CMOS and IC can drive it directly, which predigests the driving circuit and reduces cost. The grid-source part of MOSFET can be regarded as a capacitor. Commonly, the grid-driving circuit used ineludes TTL driving circuit. Fig. 12. 13 shows a TTL driving circuit with emitter follower. 5V
Fig. 12. 13
15V
TTL driving circuit with emitter follower
According to different topological structures, the power amplifier circuit made of MOSFET has three types- 22 - 23J •
1. Push-pull converter Figure 12. l1(a) is the main circuit of a push-pull converter, which is excited by the driving signals applied to the gates, two switching elements conduct alternately through the middle point of the primary side of the transformer. Fig. 12. 14(b) shows the waveforms on every node. We can sec from this figure that the unipolar square waves alternately applied to the gate arc converted to bipolar square waves by this circuit. Due to the very low conducting resistance and leak current, the loss of MOSFET is very small in a period from ON to OFF. Q, Grid voltage
nsLJ
LnJL 0, Drain voltage LnJL O, Gridvollage
Q2Drain voltage Outp ut ,'oltage
Cal Main circuit
nsLJ
oor uu
(b) Wavefonns of nodes
Fig. 12. 14
Push-pull converter
Chapter 12
361
Driving Techniques for Ultrasonic Motors
In the push-pull circuit and the following full-bridge and half-bridge circuits, it is not allowed that the series of two switehing MOSFETs conduct simultaneously, whieh will eause damage of the MOSFETs. Due to this reason, we have to
use driving signals with the duty cycle of less than 50%. This means that the dead-zone is indispensable, see section 12. 3. Push-pull converter is the simplest structure for power amplifier, suitable for the driver using low-voltage DC supply.
2. Full bridge converter In push-pull converter, the rating voltage of the MOSFET is at least two times of DC supply. For safety design, the rating voltage of the MOSFET should be 3. 3 times of the supply. If DC power is supplied by rectification of AC network, the voltage on the MOSFET may be 1 OOOV in worst condition. At present, the switching MOSFET of rating voltage 1 OOOV with suitable switching speed is very expensive. Therefore, we seldom use push-pull circuit when the source is directly supplied by AC network.
II[
Vout
Vout
+--------,*
Fig. 12. 15
Full bridge converter
+-----------,11 [
t;fL Fig. 12. 16
Half bridge converter
However, a full-bridge converter can solve this problem. Fig. 12. 15 is the circuit of the full-bridge converter. The opposite MOSFETs QJ and Q" Q2 and ~conduct alternately, namely, the QJ and Q, conduct within the first half period, Q2and Q4 conduct within the second half period. We can see from this figure that the voltage on the MOSFET is half of that of push-pull converter. The reliability of the full-bridge is enhanced, but four MOSFETs are used. Therefore, the full-bridge converter is suitable for power supply with high voltage.
3. Half bridge converter One arm of the full-bridge can be substituted by two capacitors, as shown in Fig. 12. 16. This is a half-bridge converter, which is widely used in low power applications. The voltage of the middle point of capacitors is about V /2, so the primary voltage of the transformer is V /2, and that of full-bridge is V. This means that for same power, the primary current of the half-bridge is double of that of the full-bridge.
362
12. 5 12. 5. 1
Ultrasonic Motors Technologies and Ap plicalions
Electrical Characteristics of Ultrasonic Motors Experimental Results and System Description
1. Experimental phenomena The stator made of a piezoelectric element and a metallic elastomer is the key part of ultrasonic motors. The vibration of the stator, the friction between the stator and rotor, temperature, etc. have influence on the piezoelectric element's performance, and USM's characteristics. The electrical admittance curve is almost symmetric near the resonant frequency when the stator of USM is driven by low power, and the electric and mechanical characteristics arc almost the same, which can be explained by linear theory of piezoelectric material. However, the drive voltage applied to TRUM is usually higher than 300Vpp and output power is more than 8W. In this case, some special phenomena are shown: temperature increases obviously, resonant frequency drifts seriously as shown in Fig. 12. 17, and some nonlinear phenomena happen, such as electrical admittance curve of stator from low to high frequencies does not coincide with that from high to low frequencies,
as shown
in Fig. 12. 18. 0.0 12 , - - - - - - - - - - - - - - - - , - - - Curve rrom low 10 high frequencies 0.010 Co I'Ve fI'ol1l high to
100 __- - - Increasing lemperal\lJe
90 80
:5 E
e 0.008
70
7
60
g
1l
~ 50
i
low frequencies
"
-6
40
<
'" 30 20
--
10 0
40
40.5
41
41.5
42
42.5
Freq "tllcyik Hz
Fig. 12. 17 Relationship between speed, frequency and temperature of TRUM
0,006
0,004 0.002 O~------L-----~~----~
4,1
4. 15
4.2
11Hz
4.25 X IO"
Fig. 12. 18 Frequency response of stator under high power
The nonlinear characteristics of the stator inelude nonlinear of dielectric, elasticity, and piezoelectricity. So the linear piezoelectric equations, where corresponding coefficients are constants, cannot be used to describe the energy transformation and dissipation of the stator driven by high power. So we must reinvestigate and understand the vibrating state and driving condition of the stator. In this section some primary eonelusions arc given by analyses and testing.
2. Description and analysis of system Based on the mathematic description of nonlinear and hysteresis, and other
Chapter 12
Driving Techniques for Ultrasonic Motors
363
researchers' investigating results. the piezoelectric equations can be written as complcx form-,,5,24 26 -. Considcring thc dissipation, hystcrcsis, and nonlincarity. thc admittancc of a stator undcr high powcr can bc rcwrittcn as Y
=
iwC o (1- itanO')
+ iwCoK;j [1- i(2tanB- tan¢)]
• [tan(wl/2c e )/(wl/2ce ) ]
(12. 5)
whcrc tanO', tan¢ , and tanB dcnotc thc dissipations (or losscs) of dielcctric, elasticity, and piezoelectricity, respectively. The elamped capacitance is defined by Co = (l - k; l ) Clb / h p ) Eo E; and K;l = k;l / (1 - k;l ) • where k'l is the electromechanical coupling cocfficicnt of PZT. Thc sound spccd with dissipation is dcnotcd by c e • Thc lcngth, width, and thickncss of thc piczoelcctric elcmcnt arc cxprcsscd by l, b, and hI" rcspcctively. Thc dielcctric constant in vacuum and frcc dielcctric constant of piczoelcctric elcmcnt arc rcprcscntcd by Eo and E; (scc Chap. 2), rcs pccti vely. According to the admittance characteristic of the stator with hysteresis and dissipation under high power, the equivalent circuits at the resonant and anti-resonant point arc obtaincdL3J as shown in Figs. 12. 19 and 12. 20, rcspcctively. Thc rcsistancc, inductancc, and capacitancc arc approximately constant in low powcr case but nonlinear variables in high power case.
c, o
Fig. 12. 19
Equivalent circuit of stator near the resonant frequency
Fig. 12. 20
Equivalent circuit of stator near the anti-resonant frequeney
I
Fig. 12. 21
Improved equivalent circuit of USM
It is difficult that the accurate electrical model of USM is derived if the dissipation, friction, and heat are considered. Based on the engineering approximation and the utilization of Taylor series, an equivalent circuit can be shown in Fig. 12. 21. According to cxpcrimcntal rcsults, thc elampcd rcsistancc (R d ) and capaci tance (Co) are approximately constant. On the other hand, the impedance Zm of the dynamic branch varies sharply, that is elosely related to the vibration of
364
Ultrasonic Motors Technologies and Ap plicalions
the stator. The impedance can be described as (12. 6)
where ZmO is an approximate constant impedance for low power, and the high order term of Taylor series, LZ(Im)' is the function of the current of the dynamic branch, which is directly related to the vibration speed of the stator, so LZ(Im) is also the function of the vibration speed. Some experiments have proved that the vibration characteristics and dissipation of the stator, where the dissipation depicted by mechanical quality factor, are almost proportional to the square of vibration speed.
12. 5. 2
Analysis of Vibration States and Driving Method
Theoretical analyses show that the resonance and anti-resonance are very different vibration states of a stator 26 - 27J • In general, the largest and smallest admittance of the stator arc defined as the resonance and anti-resonance, respectively. The properties of the stator near the resonant and anti-resonant frequencies are: (1) When the driving frequency approaches resonant frequency, the impedance is small and varies sharply. In this case, the fluctuation of drive current is large, which leads to poor stability of driving voltage and current. (2) Near anti-resonant frequency, the impedance of the stator is relatively large and fluctuates little. In this case, the stability is good. (3) Near the resonant frequency, the current should be large enough to obtain considerable vibration amplitude and speed. However, large current will reduce the performance of piezoelectric element, such as high losses, bad quality factor, material aging, etc. (4) Ncar the anti-resonant frequency, large voltage is necessary to obtain certain response. The piezoelectric element can endure a high voltage. In this case, the stator shows better synthetical performance. From viewpoints of power supply, either in the resonance or anti-resonance states can meet the need of the stator's vibration. The choice of the excited states depends on some practical factors: a load, output power, efficiency, control technique, and so on. When the motor operates ncar the resonant frequency, the impedance of the stator is small, the output power of driver is higher, and the efficiency of the motor is low. On the other hand, when the motor operates near the anti-resonant frequency, the impedance of the stator is large, and the high efficiency of the motor can be obtained due to the small input power.
12. 5. 3
Experimental Results
In order to investigate the performance of the stator of USM at resonance and anti-resonance states, TRUM-60 is chosen as testing sample. Experimental results are compared with each other.
1. Resonant characteristics of the stator driven by constant voltage Frequency sweep experiments for the TRUM-60 stator arc conducted with PSV-
Chapter 12
365
Driving Techniques for Ultrasonic Motors
300F-B. Driving voltages are 10Vpp • 20Vpp , 10Vpp • 60Vpp • and SOVPP ' respectively. Their results are shown in Fig. 12. 22 (a). The current variation under the constant voltage value of SOVPP ' is shown in Fig. 12. 22(b). in which fitting curves of testing data are (1), (2), and (3) , where arrowhead means the increase tendency of driving frequency. The cross point of curves (1) and (2) is the maximum current point (the maximum admittance point). corresponding to the resonant frequency. The cross point of curves (2) and (3) is the minimum current point (the minimum admittance point), corresponding to the anti-resonant frequency. Based on the experimental results, the response characteristics, of the stator driven by constant voltage arc as follows: (1) The large amplitude of the stator can be obtained ncar the resonant frequency (the maximum admittance point), and the amplitude will increase with the increase of driving voltages, resonant frequencies arc shifted and become small, The larger the driving voltage is, the more severe the shift is. 5
.
4.5 80V - - - . 60V -- - -. 40V _. _ .. 20V - - - 10V
4
5 "i
4.0 -
3.5
§
"i
~
"'" -.@
<"
<"
0. 2
0.
Data point
:P
3.0 2.5 2,0 1.5 1.0
".
0.5 0
41
41.5
42
42.5
0
100
200
,.
,,
/.%, 300
400
500
600
700
urrcntJl11A
FrequencylkHz (a) Frequency response curves
Fig. 12. 22
0
~
,,
Fit cLlfve
(b) Response VS, excitation current under the constant voltage of 80V""
Response characteristics of the stator
(2) Comparing with curves of 10Vpp to SOVpp in Fig. 12.22 (a), response curves are almost symmetric under low power, but curves corresponding to high power are asymmetric, which denotes the nonlinear characteristic of the stator. In Fig. 12. 22(b), the curve (1) and curve (2) are nearly in parallel, which shows the drastic changes of the electric characteristic. (3) Curves of amplitude and driving current in Fig. 12. 22 (b) can be considered to be subsection a linearity, which also denotes that the effective freguency range can be divided into three different driving bands under constant voltages.
2. Anti-resonant characteristics of the stator driven by constant current Similarly. the frequency sweep experiments of the stator arc conducted with PSV-300F-B. Driving current arc 30mApp , 50mApp , 70mApp , 100mApp , and 121mApp. and their results are obtained. as shown in Fig. 12. 23. The fitting curves are (1), (2), and (3) in Fig 12. 23(b), where arrowhead means the increase tendency of driving frequency. The cross point of curves (1) and (2) IS the minimum voltage point (the maximum admittance point), corresponding to
366
Ultrasonic Motors Technologies and Ap plicalions
the resonant frequency. The cross point of curves (2) and (3) is the maximum voltage point (the minimum admittance point). corresponding to the anti-resonant frequency. Based on the experimental results, the response characteristics of the stator driven by constant current are as follows: (1) The large amplitude of the stator can be obtained ncar the anti-resonant frequency, not the resonant frequency. The amplitude will increase with the increase of driving current, the anti-resonant frequency drifts little. (2) In Fig. 12. 23 (a), the response curves are almost symmetric under the constant driving current, which means that the nonlinear characteristic of the stator is not severe. (3) Fig. 12. 23(b) shows that the effective frequency range can be also divided into three different driving bands under constant current.
4
<;.
3
]
..
4.0
-
3.5
Data point FitcLI,ve
5 3.0 -S 2.5
- ~ - ~
{
C. E
4.5
121mA --_ . 100 mA 70 mA 50mA 30 mA
""
.~ 2.0 C.
2
"
(3)
-< 1.5
-<
1.0 0.5
0 41.2
4 1.4
4 1.6
41.8
42
0
50
100
150
200
250
300
350
Voltage/V
FrequencylkHz (a) Frequency response curves
Fig. 12. 23
42.2
(b) Response v .excitation voltages under conSlilnt current value of 100 mA
Response characteristics of the stator
3. Performance comparisons Figure 12. 21 (a) shows the shifting tendency of the resonant and anti-resonant frequencies with input powers. The stator driven at the anti-resonant. where the small fluctuation can be obtained, has better stability. While the resonant frequency decreases with the increase of the input power. Furthermore, the resonant frequency is almost linear variation to the square of the amplitude of the stator, as shown in Fig. 12. 21(b). Figs. 12. 21(c) and 12. 21(d) show the changing tendency of the mechanical quality factor and electromechanical coupling coefficient of the stator ncar the resonant and anti-resonant frequencies. It is found that the same performance of the stator under low input power can be achieved ncar resonant and anti-resonant frequencies. Also. while increasing input power the mechanical resonance quality factor decreases and the electromechanical coupling coefficient increases. Generally, the synthetical performance of the stator excited near anti-resonant frequency is better than that near resonant frequency. Therefore, it is proposed that the frequency near anti-resonant frequency should be chosen as driving frequency for USM.
Chapter 12
4.21 XID' '"
4.20 1--- --_
'"
4. 19
;; <>
..5§
§" ~
Driving Techniques for Ultrasonic Motors
4. 176 rX.:.:I"O ,_' - - - - - - - - - - - - ,
Resonant frequency
-
-_. Ami-resOtlanl frequency
}
4. 172
C-
4.168
4. 166
~
4.164
~
!i1
4.18
O>
4.17
~
0>
4.16
O
I
2
4
6
4. 162 4.160 4. 158 '---_-'--_----'._ _.L-_--'-_----' 15 5 10 20 25 o Sq uare amp l i tu de/~,," ' (b) Resonant frequency and amplitude
allli·resonalll frequency 800 ,--------------------,
g 700
.rl
.f' !'i
0-
650 600
,,
• Data cOI'T"esponding to resonance o Da.ta corresponding 10 and -resonance
0028 , . . . - - - - - - - - - - - - - - , -
ij
.c;
8 OIl
-= is. 8" 1l
-"- .... ... ... ...
500
450
I
400
0>
3
4
0.026 ---
I:E 0.024
to
2
6
Fit curve Measurtd dal8
•
4. 170
Inpu t powerrw (a) Shift of resona.1I frequency and
750
-
4. 174
~'~-- ---e----~----~
4. 15
367
7
iii
ear reSOnanl fre-
quency
ear anti -res-ooanl
frequency
0.022
~ /" ~ P
--"
•
/ , /r;('
0.020
0.0 18
/ , / p'
0.016 0.014
0.012
'L.._~_~_-'-_
o
2
3
_'__
4
_'__.J
5
In put powerrw
Input powerfW
(c) Mechanical resonance quality factor
(d) Electromechanical coupling coefficieru
6
Comparison of characteristics of stator near the resonant frequency with anti-resonant one
Fig. 12. 24
In fact, the contact force between the stator and rotor cannot be ignored. Moreover, the contact force is different with various loads, so the equivalent circuit cannot completely describe the response characteristic of the stator while the motor is operating. In this case, the parameters must be experimentally obtained with an operating USM. Therefore, there are certain difficulties to design USM based on equivalent circuit.
12.6
Influence of Matching Circuit on Performance of Driver
The functions of matching circuit are: (Dpower matching: the USM is a capacitive load. power matching can reduce reactive loss and increase the efficiency of the system; CZ) filtering: the output of the transformer is high voltage bipolar square wave, which comprises many harmonic components. A matching circuit could filter the unnecessary harmonic waves to achieve a necessary and basic wave. avoiding exciting non-operating modes of a stator- 3 • 26. 28J. To reduce the self-lose of the matching circuit, it is often composed of capacitor or inductor. The matching circuit inevitably influences the electrical and vi-
368
Ultrasonic Motors Technologies and Ap plicalions
bration characteristics of USM. Therefore, study on the matching circuit is very important to design the driver. According to the simplified equivalent circuit in Fig 12.21 and the impedance expression of the stator, the equivalent impedance of the stator can be written as following when the dynamic resistor RHO and the parallel resistor Rd are omitted for simplicity
Z
=
iX
= _
i _1_ • 1 - ( f j f) 2 we o l-(fp/f)'
02. 7)
where Co is the static elamped capacitance of the motor, I, and II' are the series resonance and parallel resonance frequencies of the stator, respectively. Although there are many kinds of matching circuits, it is usually composed of inductors or capacitors. Here we will take the stator of a disk-type USM as an object to investigate, PSV-300F-B used as testing tool to study the influence of the matching inductor and capacitor on the electrical and vibration performance.
12. 6. 1
Influence of Matching Capacitor
1. Series capacitor When the stator is in series with eapaeitor C, the equivalent impedanee is
Z'
=-
i[---.L + _1_ • 1- ( f j f): ] we
we o
1 - (fe/f)
02. 8)
For electrical analysis, the matching capacitor is chosen as 30nF, o. lll-F, O.18Il-F, 1. 0ll-F, 3. 31l-F, l0ll-F, and 171l-F. Fig. 12. 25 shows the sweep results, where Fig. 12. 25(a) is for the former four eapacitors, and Fig. 12. 25(b) is for the latter four capacitors. It is shown from testing results that for the stator in series with capacitor, the resonance frequency of the stator I, (""" I,) rises, and the anti-resonance frequency I, (""" Ip) keeps the same. Besides, the less the series capacitor is, the higher the resonance frequency is, and the less the equivalent admittance is. The series capacitor translates the phase frequency curve of the motor near the resonance frequency. But when the series capacitor is greater than certain value 00ll-F in this case) , the amplitude and phase frequencies characteristic keeps the same, as shown in Fig. 12. 25(b). At the same time, when in series with capacitor the vibration characteristic of the stator is measured with PSV-300F-B, applying constant voltage value of 20Vpp • The vibration characteristic of the stator is elose to that of non-matching in series with larger capacitance. But with decreasing the capacitance, the vibration amplitude becomes smaller, and the resonance frequency is higher. Fig. 12.26 shows the vibration characteristic for the capacitances of 30nF and lOnF. It can be seen from this figure that the frequency change of vibration characteristic is coincident with that of electrical characteristic. Smaller capacitance makes the equivalent impedance larger, which reduces the driving current and then the amplitude.
Chapter 12
0.02
100
0,015
50
Non-matching
~
§
,S:
""~
0.01
10 ~I'
0
5:
.§ ..:
369
Driving Techniques 10r Ultrasonic Motors
- 50
0,005
0 4, 15 4,16
4,17
4,18
4, 19
42 , 4.21
4.22
1'Hz
x lO'
-100 L-_L-_'--_'--_'--_''_____-'-------' 4,154. 16 4.174, 184. 19 4 .2 4214.22 . x lO' 11Hz
(a) Freqllcncy sweep cllrve I
0,02 50
0.0 15 (/)
]
"'1;l"
~ 0,0 1
"§
0
~
c::
"0
..:
-50
0.005
o
-I OO ~_'___"_____"_____"_____-'----_~~
4. 15 4 . 16
4.17
4. 18 4. 19 [ 1Hz
4 .2
421 .
4,22 xlD'
4. 15
4. 16
4 .17 4,18 4 . 19 [fl-lz
4.2
421. 4.22 xlD'
(b) Frequency sweep curve 2
Fig. 12. 25
Testing characteristics of stator with series capacitor 1.8 ,------,,------,-----y--,,----,.----,.-----, 1.6
14
0.8 0.6
0.4 0.2 0
4,14
4. 12
4 ,16
4, 18
4,2
4.22
[fl-lz
Fig. 12.26
4.24 xlO'
Amplitude-frequency characteristics 01 the stator with series capacitor
2. Parallel capacitor When the stator is in parallel with capacitor C, the equivalent admittance is Y
I
.
=
{
wC
+ wC o •
1- (fJf)' 1 - ( f j f) 2
]
(12. 9)
Ultrasonic Motors Technologies and Ap plicalions
370
,..
0.04 r----,.--r---r----,,----r--y---.--........
0.03 ~
__ - /
...
,
I
I I
I1
0.1!1F
I
,
.. _ _ - - - - -
"
!I ~0.02
i<
...... -..
0.01
o
4.lS 4.16 4.17 4.18 4.19 4.2
4.21 4.22 4.23
JIHz Fig. 12.27
-100 L...--'-----L_....L...---'-----''--.........---L.----J 4.\S 4.16
4.17 4.18
4.19 4.2
4.21
JIHz
x10"
4.22 4.23
xlO'
Electrical characteristics of stator with parallel capacitor
By using the similar methods of series capacitor, we make the capacitances of 30nF, O.l,uF, O. 48,uF, 1. O,uF, 3. 3,uF, 10,uF, and 47,uF in parallel with the stator. Fig. 12.27 shows the electrical characteristics for 30nF and O. 1,uF. In Doppler laser vibration measurement, capacitances of 30nF, O. 1,uF, and O. 18,uF are in parallel with stator to make the vibration measurement applying constant voltage value of 20V PI" Fig. 12. 28 shows results of the parallel capacitor. By comparing the Fig. 12. 27 with 12. 28, we can conelude that: (1) For parallel capacitor, the resonance frequency of the stator Ie keeps the same, and the anti-resonance frequency I, decreases. the larger the capacitance is, the less I, is. (2) Compared with non-matching, the change of the vibration amplitude of larger parallel capacitance is very small, but the total current increases dramatically. The reason is that the larger capacitance reduces the total impedance, which makes the input current very large. But the parallel capacitor shunts the added current, and the current feed into the motor is invariable, so the excited vibration amplitude keeps the same.
....
2~--~---r--~----~--~--
I.S
i
]
:a ~
o.s 0
~~
~~
~~
-o.s L.-_ _...1..-_ _....L.._ _----'_ _ _ _-'--_ _- ' -_ _---' 4.12
4.14
4.16
4.18
flH2
Fig. 12.28
4.2
4.22
4.24 ><10'
Vibration characteristics of stator with parallel capacitor
Chapter 12
12.6.2
371
Driving Techniques for Ultrasonic Motors
Influence of Matching Inductors
1. Series inductor When the stator is in series with inductor L, the equivalent impedance is Z
, .[L =
1 w
1
we o
-
•
1 - ( f j f)' ]
1-
(12. 10)
CfJn'
For electrical analysis, the matching inductances arc chosen as 1. OmH, 1. 2mH, 1. 5mH, and 2. OmH. Fig. 12. 29 shows testing results. At the same time, the vibration characteristics of the stator which is in series with inductance of 1. OmH to 1. 2mH arc measured using PSV-300F-B applying constant voltage value of 20VI'I" Fig. 12. 30 shows testing results. 100 , - - - . - - - . - - . - - - . - - . - - - ,
0.08 , - - - - , - - - , - - - - , - - - , - - - , - - - ,
0,06
'"'O!!J 0,05
I
I
10
i ~~/ ' .,1,
I
0.01
/
1.0mH
2.0mH
4 .1
:j
i~
: ;! " , .'., \, \~.,
- 50
1.'\ ,
2.0 mH
\~~.
\"
o~-~-~~~~--~--~-~
4
I
1,5 mH : ~; --.....
m
!~: ~
...~~"
\ \\
'k :i
~i
H
/ Lt Non-matching J
0,02
1,_mH
'10'"
I Ii i I I ..
, '"
0.03
')
50
"
.~ 0,04
~
"<-"":'.
[V I .5 I1lH " . ~ ~ Tt./ 1.2 mH
0.07
4 ,2
4,3
4 ,4
4,5
4,6
11Hz
Fig. 12.29
- IOO~-~--~-~~-~--~-~
4
4, 1
4,2
x i O'
4.3
4.4
/1Hz
4,5
4,6
xU)'
Electrical characteristics of stator with series inductor 2.0 ,----.-.-----,,----,----:-....-----,.------,--....--n 1.8
1.2 mH :'
1.6
~
,/ . \!
E 1.4
i
1.2
]
I
1.5 mH
< 0.6
0.2 O
~ ,
~/ /\ , . ,, ,,
to.8 0.4
I I
--_ ... -'
i' .,"
L-~_~~L--L_~~L-_L_~_U
4.06 4.08 4.1 4. 12 4.14 4. 16 4. 18 4.2 4.22 4.24
[1Hz
Fig. 12.30
x lO'
Vibration characteristics of stator with series inductor
It can be seen from Fig. 12. 29 and 12. 30 that when the stator is in series with inductor: (1) The resonance frequency f, drops and the anti-resonance frequency f, keeps the same. A larger inductance makes f, lower in certain scope. The series
372
Ultrasonic Motors Technologies and Ap plicalions
inductor translates the phase frequency curve of the stator near the resonance frequency. (2) The series inductor will create a new resonance frequency beyond the antiresonance frequency theoretically. When the inductance is very small, this new resonance frequency is far greater than fe' However, when the inductance is rather large (2. OmH in this case), this new resonance frequency will be very close to f, and f" which will affect the stator's electrical and vibration characteristics. (3) For series inductor, the frequency change for vibration characteristic is coincident with that for electrical characteristic. A larger inductance makes the equivalent impedance smaller, which increases the driving current and then the vibration amplitude.
2. Parallel inductor When the stator is in parallel with inductor L, the equivalent admittance is Y'
=
'X'
=
1
,[1
_1 wL
+ w c . 1CIt'! f)2 ] 1 - (fJn' 0
(12. 11)
By using the similar methods for series inductor, inductances can be made of
1. OmH, 1. 2mH, 1. 5mH, and 2. OmH in parallel with the stator. Figs. 12. 31 and 12.32 show testing results, where Fig. 12. 32 is obtained applying constant voltage of 20V pp • By comparing the Figs. 12.31 and 12. 32, we can obtain following conclusions: (1) For parallel inductor, the resonance frequency anti resonance frequency
f, keeps the same, but the
f, rises. The smaller the parallel inductance is, the
higher fe is (in this case, when the parallel inductance is less than 2. OmH, the fe is beyond the scope of the vibrometer).
(2) A new anti-resonance frequency occurs at the frequency lower than
f,.
The smaller the parallel inductance is, the higher this new anti-resonance frequency is. The phase-frequency curves of the stator at this frequency change dramati cally , and the frequency band is very narrow. (3) Since the voltage applied to the stator is the same, the vibration characteristics of the stator for both parallel inductor and non-matching are almost the same. From the above analysis for capacitor and inductor matching circuits, we can draw conclusions as follows: (1) The core of matching circuit is to reduce the loss of the driver, and let a stator vibrate more efficiently. The series capacitor or inductor mainly affects the characteristics near the resonance frequency, and the parallel capacitor or inductor mainly affects the characteristics near the anti-resonance frequency. Therefore, before selecting suitable matching circuit, we should choose proper operating frequency range according to the requirement of the motor's power. (2) In the condition of constant voltage driving, the in series inductor matching will lead to a higher voltage on the motor due to the resonance, and the
Chapter 12
Driving Techniques for Ultrasonic Motors
373
voltage applied to the motor is sensitive to the load, so it is an important factor to causc thc change of vibration characteristics. On the other hand, the parallel matching is very different. Compared with series matching, the vibration characteristics are relatively stable due to the applied voltage value is constant. (3) On the precondition of lesser influence to the vibration of the stator, suitable series capacitor could improve performances of USM. Since USM is capacitive load, the inductor is indispensable in the matching circuit, but it have to be selected very carefully. 0.02 ,...--,---r--,..--,---r--,..---,.---,---,
100 ....---.----,---.--.---.--,---,..--,--,
: 0.015
on-matching
:V
50
f!2 1l ~ 0.0 1
'"
1.0 mH
-i3
<: 000 - NOll -matching: .
~
) :"
-50
4.3 4 3. 5 4.4
11Hz Fig. 12.31
4.45 4.5 xlO'
- IOO ~~~~~~~~~~~~~~~
4.05 4.1
4.15 4 .2
425 . 4.3 4 .35 4.4 4.45 4.5 x lO' JIHz
Electrical characteristics of stator with parallel inductor 2 1.2 mH
1.5 E
~
".=
~
0.5
E
<:
Non-matching
0 - 0.5
4. 12
4.14
4. 16
4. 18
4.2
JIHz
Fig. 12.32
12. 6. 3
4.22
4.24
4.26 x lO'
Vibration characteristics of stator with parallel inductor
Influence of USM on Driver
As shown in Fig. 12. 1, a fundamental signal is converted to a constant high voltage square wave through power amplifier and transformers. This square wave is filtered by LC resonance network and then applied to USM. The operating frequency is usually a little higher than the resonance frequency and more ncar to the anti-resonance frequency. Fig. 12. 33 shows the simplified model, which is the common model of USM driver.
Ultrasonic Motors Technologies and Ap plicalions
374
LC resonant circuit
---------------------1
I
Square wave
J1J1 1
Fig. 12. 33
Ultrasonic motor
.--------------,
L
C
Co
R
Simplified model of USM driver
The USM can be regarded as a capacitor in parallel with a resistor when it operates between the resonance and anti-resonance frequencies. At the same time, USM is in resonance with the LC matching circuit. Therefore, as the load of the driver, USM will influence the driver as follows: (1) The power of USM is controlled by the fundamental wave of the square wa ve, which is extracted by the resonance of LC matching circuit together with USM, and certain voltage gain is achieved at the same time when the operating frequency is fixed, the variation of the load will affect the voltage waveform and amplitude between node A and B in Fig. 12. 33. (2) The natural frequency of LC network and USM shown in Fig. 12. 33 is approximate
w =
(l//LC,,) j1 - eL/ R' C" ) , and its quality factor is Qm
=
R/
= C + Co. To get a pure sine wave, Qm should be bigger enough. However, the performance of USM is very sensitive to big Qm' Therefore it is an inconsistency, which makes the difficulty to design the matching circuit. (3) The total capacitance of the circuit C" is variable, which leads to the shifting of a natural frequency. The relative variation of the fundamental harmonic (LV/V) is strongly affected by the value of fd/f, (fd is a driving frequency) and
jL/C" ' where C"
LC"/C,, . Due to the operating principle of USM and its structural characteristic, the variation of electrical performance is inevitable. Therefore, how to reduce the influence of fluctuating of a motor's operating state on the driver is very important to keep the operating stability of USM. Currently, a common method used is the frequency automatically tracking, and good results arc already achieved. However, the frequency band of USM is very narrow, which means the tracking precision of this method must be enough. Otherwise, the severely fluctuating frequency leads to USM operating unstability. Figure 12. 34 is the circuit diagram of open-loop USM driver developed by PDLab. It is mainly composed of five parts: a signal generator, frequency divider and phase splitter, dead-zone, power amplifier, and matching circuit. Fig. 12. 35 is the picture of the driver. One point should be mentioned that this driver could drive 4-phase USM. In this case, the transformers in Fig. 12. 34 arc replaced by the transformers with center-tap, and two series inductors arc added, shown in Fig. 12. 36. It is the same for half-bridge and full-bridge converters.
till)
Fig. 12.34
' lrc: :~:
IC \'50
1m 111\
Ul
CCM
ISS ~
I'a;
LIN
so
'"m
UM D·2 type driver's circuit developed by PDLab
' 11)\'
~
COl Ul
I'a;
l-SS
IS
,. '" U'i
'"
111\
""
15\
t(~
1I
~_
()
"" '"
CD
...rn
S-o
~
0'
~
rn 0
rt
...po
S
...0
.....,
.0 C (J rn
2,
::r
n
(J
>-J
C/Q
:::;S'
...tJ
N
f-'
rt (J
"...
::r po
376
Ultrasonic Motors Technologies and Ap plicalions
...f1..f'L 1__ Q,
M,
+v0 - - - -----+---c::-<: 1 1.*
.
~
~
.'-; ••-sinWl
+v L,
Fig. 12. 36 Power amplifier and matching of 1-phase driver
Fig. 12. 3S Picture of driver for ultrasonic motor
12.7
-cos~"
Non-transformer Driver with Resonance Voltage Step-up
Generally, USM needs a high voltage of at least 100V!'!' to be driven, most eurrent drivers are set up by push-pull transformer. However, the existence of the transformer leads to the big volume of the driver, and makes a big obstaele to the promotion of such kind of USM. Since USM is a capacitive load, a booster using LC resonance will get the required high voltage, which reduces the driver's volume greatly and fulfils the demand for general engineering: 2 , 29J. :'\lear the resonance frequency, the ultrasonic motor can be represented by an equivalent circuit of Z, as shown in Fig. 12. 37, where Co is the static capacitance of PZT, Lm is the equivalent inductance, C m and Rm are capacitance, resistance related to the stator's mass, elasticity and mechanical lose, respectively, which construct a dynamic branch. The values of these parameters are determined not only by the stator itself, but also by the pre-pressure of the rotor acting on the stator. The external impedance of the motor ZI is (12. 12) And then the equivalent impedance Z of USM in series with inductance is Z
Step up ratio is defined as Av
=
1
=
i
iwL +ZI
(12. 13)
I·
Then we can get the relationship between step-up ratio and operating frequency, as shown in Fig. 12. 38. It can be seen from this figure that the Av is very big near the resonance frequency. This means the voltage on the motor can be boos-
Chapter 12
377
Driving Techniques for Ultrasonic Motors
ted through LC resonance by using a series inductor.
70 L,.
60 50
...: 40
G.
z
z,
C"
30 20
R1n
10 °30
35
40
60
f lkHz
Fig. 12. 37 Resonance circuit of inductor with USM
Fig. 12.38 Resonance circuit step-up ratio vs. frequency
In the condition of not changing the external characteristics of the circuit, the equivalent circuit of USM can be transformed to a RC parallel circuit, as shown in Fig. 12. 39. The transformation relation of the parameters is derived as
,
1
Lm = Lm - ----'--C W
C
~ =
R
=
C
R'm
~o -
R
tn
+
(
m
L:n + (W L')
2
(12. 11)
Tn
,
WLm
) 2
Rm
Co
Fig. 12.39
Conversion of equivalent circuit for USM
The main booster circuit using LC resonance technique is shown in Fig. 12.10, where L is a series inductor, K is a switch. Diode is an ultra-fast recovery diode. and USM is represented by RC parallel circuit. The operating principle of this circuit is explained in detail as follows: when the switch K turns on, DC power E transmits energy into the L; when the switch turns off, the inductor L deliveries its energy into C and resonates with C. The whole circuit forms a LC series resonance circuit with initial condition. In Fig. 12.10, the switch K is MOSFET that can bear high voltage. Since the MOSFET has a body diode anti-paralleled with it, if without Diode, the capacitor C will discharge through this body diode when its voltage inverses, which makes the output voltage only half wave. In order to get entire oscillation with high output voltage, as shown in Fig. 12.41, an ultra-fast recovery diode Diode is in series with MOSFET. Assume the resonance between Land C is just one
Ultrasonic Motors Technologies and Ap plicalions
378
cycle during the switch OFF duration, the inductance L can be estimated by L
=
(1 (2rr) - D)' "d 0 f t h e SWltc "h"mg signa, " I an d D"IS t h e d uty cy2 CT2 ,w h cre T"IS t h e peno cle. 8
-
+
Diode
V,
@) ~--
signal
R
C
.----1
0
"0
r----1
r--1
-2 -4
V
- 60
Fig. 12.40
-K
~wi lchi~ [
2
et)
>
50 kHz
'"'
....> " r;
"~ V,
volta ge
4
r oj
~ Outpu t
~
6
10
V
V
V 20
30
40 IIIl S
50
60
70
80
Fig. 12. 41
Waveforms of LC resonance voltage step-up circuit
Primary driving
circuit without transformer
Thc detail cd operating proccss is as follows: Suppose that the waveform of the switch K is shown in Fig. 12.42, whcn t= 0, the switch turns on; at t = to, the switch turns of[; and at t = t\ it turns on again. The on-of[ cycle T= t, - to, and the duty cycle D = (t, - t 1 ) IT. Stagc I (Swi tch turns on): At t = 0, thc switch turns on, the original circuit can bc transformed into Fig. 12. 13. The C is shorted by the switch, and the power E and L make up a loop through the switch. The current i increases linearly: i = io EtlL When thc currcnt is not continuous. i = EtlL Stagc II (Swi tch turns off): At t = to, the switch K turns of[, Land C forms a resonance circuit, the original circuitcan be transformed to Fig. 12.11. The corresponding state equations are
+
J ~~' 1I C
~
K
di dt
i-~
= =
E
-
(12.15)
u,
L
E ()
10
II
C
E
R
12
Fig. 12.42 Waveform of switch signal
Fig. 12.43
Equivalent circuit with closed switch
Fig. 12.44
Equivalent circuit with open switch
Chapter 12
Driving Techniques for Ultrasonic Motors
379
When the current is discontinuous, the initial conditions of these equations are i(t = EDT/Land U, (t = 0, and assume the resistance R is very large, then the voltage u,(t) and the current i(t)at the resonant stage can be derived as j
)
j
{
h were w Let
~'
(t)
=
E [1 - cosw(t - to)
i(t)
=
ECw[sinw(t- to)
Uc
(1 _2rcD) l'
=
=
)
=
1. ;rc
IS
+ DTwsinw(t -
to)]
+ DTwcosw(t- to)]
02.16)
t h e resonance angu Iar f requency.
0, then the maximum value of u,. is
=
U,.m"x
[u,(tHn"x
= E[ 1 + /1 + (DTw)']
02. 17)
Thc above analyses are derived from the assumption that Land C only resonate one cycle at the OFF stage. In fact, we can choose different L so as to let them resonate k (k = 1 ,2,3 ... ) cycles at the OFF stage, and the inductance should be decided by
L=
O-D)'T' (2rcS)'C
02. 18)
2rcS -:-O-:---....:.D=-:-)=1'
02.19)
And the angular frequency is w -
Substitute Eq. 02. 19) to 02. 17)
02. 20) Then we can get the rclations between maximum voltage unn"x and duty cycle under different resonance cycles k, as shown in Fig. 12. 45. It can be seen from this figure that the maximum voltage increases with the increase of k under the 20 ····1 cycle - 2 cycle --- 3 cycle
18 16
~
OIl
lS "0 >
E E
" . ~
:2
I
14
I
I
I
I
I
I
I
I
I
///
12 10 8 6 4 .................... - .................... 2~
Q1
-- --
-----------//
......
__ __ __ __ __ __ ____ Q2 Q3 Q4 Q5 Q6 Q7 Q8 ~
~
~
~
~
~
~~
Q9
Duty cycle
Fig. 12. 45
Maximum voltage at different cycle wave
Ultrasonic Motors Technologies and Ap plicalions
380
same duty cycle. Therefore, we should choose high k to get sufficient high output voltage. And at the same k, the U cmox increases with the increase of duty cycle, which is easy to understand. Larger duty cycle leads to more energy stored in the inductor, which causes a higher voltage during resonance duration. However, the voltage applied to USM is not an ideal sine wave, as shown in Fig. 12.41, which contains abundant harmonic waves. In order to drive USM efficiently, we should choose suitable harmonic wave. Therefore, it's necessary to analyze the output voltage using Fourier analyses. According to Fourier transformation, the output voltage can be rewritten as
k=]
let n
=
wT z; ,m =
ak =
k=]
DTw ,a
=
(1- D)21(, then the parameters in Eq. (12.21) are
~f2rr Uccoskx dx 0
1(
E{ 1 . k _sin(n+k)a_sin(n-k)a sm a 2(n+k) 2(n-k)
-; k
bk
+ m[1-cos(n+k)a+1-cos(n-k)a]} 2 n+k n-k 1 f,rr U c sinkx dx 1(
E 1(
{1- cosk a _ 1- cos(n+ k)a 0
k
1 - eos(n - k)a
2(n+k)
2(n-k)
+ m [Sin(n+k)a _ sin(n-k)a]} 2 n+k n-k
Therefore, the amplitudes of every harmonic wave under different duty cycle can be derived, as shown in Figs. 12.16 and 12. 17.
-c\ 5 4
- C2
Shutoff period is 1 cycle
.... C3 --- C4
_.- C,
2
/ I
""
I
I
I
I
I
I
I
.=:-;: .... ..:....~><';7 ........
O~~~~~~~~-~--L-~~
0.1
0.2
0.3
0.4
0.5 0.6 Duty cycle
0.7
0.8
0.9
Fig. 12. 46 Relationship between harmonic amplitude and duty cycle (1 cycle during shutoff)
Chapter 12
Driving Techniques for Ultrasonic Motors
381
8,--,---.---,---.---.--,---.---. _ C Shutoff period is 2 cycles.'·-. ___ e21 ,/ '\
7 6
i
.... C3
\.
'.
- C4
_.- C5
.. 3 :~~,.;",--
2
.....
.}I
/
o .-
/
.<"
0.1
0.2
0.3
0.4
0.5 0.6 Duty cycle
0.7
0.8
0.9
Relationship between harmonic amplitude and duty cycle (2 cycles during shutoff)
Fig. 12. 47
It can be seen from the above figures that different resonance cycles during OFF stage leads to different amplitudes of harmonic values. For choosing suitable circuit parameters, further analysis on the output voltage is required. Since harmonic components are very complex, their proportions among the output voltages are very different. However, only the harmonic component, whose frequency equals to USM operating frequency, can drive USM, other harmonics are consumed by heat. In order to select appropriate harmonic component to drive USM efficiently, we should know the relations of energy between every harmonic component and output voltage. The energy of the output voltage can be written as W =
f:
u;dx =
+
3 m' ---2-a
J:
(1- cosnx
+ DTwsinnx)2dx
1 - m' . + --4---s1112na n
2. -Slllna n
+ -2m (1 n
cosna)
m + -2 (cos2na n
1)
(12. 22) and that of the harmonic component is
Wk
=
[J; (x)dx
=
(a;
+ b;h
(12. 23)
Therefore, the energy efficiency R can be defined as
R
W
=
Wk
W
(12. 24)
Figs. 12.48 and 12. 49 are the rclations between Rand D under different resonance cycles. These figures indicate that when the resonance cycle is 1 during OFF stage, the 2nd harmonic component with D= 0.4 is suitable to drive USM, in this case the amplitude and energy efficiency is relatively high. While when the k=2, we should choose the 3rd harmonic with D=O. 3, in this case the energy efficiency is the highest. Of course, final parameters should be determined according to prac-
Ultrasonic Motors Technologies and Ap plicalions
382
0.7 ,----,,--,--,--,---,---,---,---, 0.6 0.5
- w, -
W,
Sh ulolT period is I cycle
·····W,
---W. _.- W,
~ 0.4 ~~
'"
0.3 0.2 0. 1 O .c~~~~~~~~~~L-~~~
0. 1 0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
DUly cycle
Fig. 12. 48 Energy efficiency vs. duty cycle (1 cycles during shutoff) 0.8 Shutoff period is 2 cycles -
0.7
~
~-
'"
11',
-.- W,
0.6
:..
0.5
..
- w,
.... ..... ~;.'\
., , ,. , '' ,, ,, , ,, ,, ,, , I',
0.4
/
0.3 0.2 0. 1
, 0.3
0.4
,
'-
0.5
. ... W. --- W,
~
\
',
0.6
"
0.7
0.8
0.9
Duly cycle
Fig. 12. 49 Energy efficiency vs. duty cycle (2 cycles during shutoff)
Fig. 12. SO
Driving device without transformer developed by PDLab
tical situation. To verify the rationality of the proposed scheme. an ultrasonic motor with diameter of 5mm is used for a sample. Its static capacitance is
o.
56nF, driving
frequency is about 75kHz, and couple inductance L is selected as 7. 2mH. The
Chapter 12
383
Driving Techniques 10r Ultrasonic Motors
whole system and experimental results are shown in Figs. 12. 50-12. 52. The amplitude of the output voltage can reach 200Vpp , the motor operates well, which proves this scheme is valid and feasible. However, the output voltage is not pure sme wave, which contains many harmonic components, this causes low efficiency when driving a high power USM. Therefore, this scheme is only suitable for low power USM .
.... ...
......; .......... 1··············· .. ··f··············
'
.. .....
.,
."""~ 2 .................. ,....................... . :
>
.,., 0
I
....................................... 0
V 5/1 divide
5/1s/di vide
Fig. 12. 51 Practical driving waveform (l cycle during shutoIO
Fig. 12. 52 Practical driving waveform (2 cycles during shuto1f)
References [ 1
J
Chunsheng Zhao. Recent progress in ultrasonic motor techniques. Measurement
J oumal
of Vibration,
& Diagnosis, 2001, 21 (1): 1-5. (in Chinese)
J
Huafcng Li. Research on Ultrasonic Motor Driver. Post-doctoral Report. Nanjing: Nanjing
[ 3J
liakui Zu. Research on Driving and Control Techniques for Traveling- Wave Ultrasonic Mo-
[ 2
University of Aeronautics and Astronautics, 2004. (in Chinese) tor Based on Its Electric Characteristics. Post-doctoral Report. :'-Ianjing: :'-Ianjing University
of Aeronautics and Astronautics, 2001. (in Chinese)
J
Zhihua Chen. Research on the Control of Ultrasonic Motors. Post-doctoral Report. Nanjing:
[ 5J
H uafcng Li. Study on Ultrasonic Motor and Its Precise Servo-control System. Dissertation
[ 1
:'-Ianjing University of Aeronautics and Astronautics, 2003. (in Chinese) for the Degree of Doctor of Philosophy. Wuhan: Huazhong University of Science and Technology, 2002. (in Chinese) [ 6
J
Shoushui Wei. Drive and Control Technology of Ultrasonic Motors. Dissertation for the Degree of Doctor of Philosophy. Nanjing: :'-Ianjing University of Aeronautics and Astronautics, 2000. (in Chinese)
[ 7
J
Chunsheng Zhao. Some proposals for development of ultrasonic motor techniques in China.
[ 8
J
Dongliang Xu. Research on
Micromotors Servo Technique, 2006, 39(2): 64-67. (in Chinese) Self~oscillation
Driver of Ultrasonic Motors. Dissertation for the
Degree of Master. :'-Ianjing: :'-Ianjing University of Aeronautics and Astronautics, 2005. (in Chinese)
J
Hyeoung Woo Kim, Shuxiang Dong, Pitak Laoratanakul. et al. :'-Iovel method for driving the
[10J
Chunsheng Zhao, Bin Zhou. Micro driver for four-phase ultrasonic motor. National Inven-
[ 9
ultrasonic motor. IEEE Trans. on UFFC, 2002(10): 1356-1362.
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tion Patent, ZL02138534. 3,2002-11-15. (in Chinese)
[llJ
Jian-Shiang Chen, In-Dar Lin. Toward the implementation of an ultrasonic motor servo drive
[12J
Suwan Manuspiya, Pitak Laoratanakul, Kenji Uehino. Integration o[ a piezoelectric trans-
using FPCA. Mechatronics, 2002, 12: 511-524. former and an ultrasonic motor. Ultrasonics, 2003,11(2): 83-87.
[l3J
Erdal Bekiroglu, :'-lihat Dalda!. Remote control of an ultrasonic motor by using a GSM mobile
[l1J
M Flueekiger, M Bullo, Y Perriard. Sensorless speed control of traveling wave ultrasonic
phone. Sensors and Actuators 11., 2005 (120): 536-542. motor. Conference Record of the 2006 IEEE Industry Applications Conference Forty-First lAS Annual Meeting. Piscataway: Institute o[ electrical and electronics engineers Inc., 2006:
5-13.
[l5J
Bal Gungor, Bekiroglu Erda!. A PWM technique for DSP controlled ultrasonic motor drive system. Electric Power Components and Systems, 2005, 33(1): 21-38.
[l6J
Jian Xu, Grant Edward, Kingon Angus I. Drive circuit for a mode conversion rotary ultrasonic motor. 31st Annual Conference of IEEE Industrial Electronics Society. Institute o[ Electrical and Electronics Engineers Computer Society, 2005: 1588-1592.
[l7J
Youguang Li, Zaili Chen. Phase shifted ZVT-PWM high-frequency full-bridge inverter with auxiliary resonance nets [or driving ultrasonic molor. Pruceedings of the Eighth Internatiun-
al Conference on Electrical Machines and Systems, 2005, 3: 1847-1850.
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Jiakui Xie, Yueqing Xuan. Electronic Circuit (Nonlinear Part). Beijing: Higher Education Press, 1988. (in Chinese)
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Huaguang Kang. Fundamentals of Electronic Technique. Beijing: Higher Education Press, 2000. (in Chinese)
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Wanjie Song, Feng Luo, Shunjun Wu. Technology and Application of CPLD. Xi'an: Press o[ Xidian University, 1999. (in Chinese)
[21J
Huafeng Li, Chunsheng Zhao. Mini-driver based on CPLD for ultrasonic motor. Proceedings of the 2005 IEEE International Ultrasonics, Ferroelectrics, and Frequency Control 51th Anniversary] oint Cunference. Piscataway: Institute of Electrical and Electronics Engineers
Inc. , 2005: 1542-1545. [22J
Changyang Hu. Class D and E Switch-mode Power Amplifier. Beijing: Higher Education Press, 1985. (inChinese)
[23J
Zhansong Zhang, Xuansan Cai. Principle and Design of Switching Power. Beijing: Publishing House of Electronics Industry, 1999. (in Chinese)
[21J
Jiakui Zu, Chunsheng Zhao. Drive modes of stator of traveling wave ultrasonic motors and its
[25J
Jiakui Zu, Chunsheng Zhao. Development of driving and control technique for traveling wave
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Chunsheng Zhao, liakui Zu. Research on resonance and antiresonanee states o[ [ree stator o[
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Huafcng Li, Chunsheng Zhao. Research on the ultrasonic motor driver based on LC resonance. Proceedings of the CSEE, 2005, 25(23): 111-118. (in Chinese)
Chapter 13
Control Techniques for Ultrasonic Motors USM's performance is influenced by many factors. Although USMs inherently possess excellent control performance, it needs appropriate control strategies to implement. Therefore, control techniques directly influence the implementation of USM's performance and furthermore will influence its applications and popularization: ll . Due to the special structure and operating principle of USMs, its output characteristics vary according to temperature, abrasion, pre-pressure, and exciting frequency, which makes it difficult to operate well under the open loop condition. It must also be formed as a elosed loop system to fully implement its potential performance. In addition, under practical operations we need to control speed, position, and torque of USMs. Therefore, the objective of USM's control is to overcome its own imperfection, improve its output characteristics, and implement its inherent excellent performance. In order to achieve these purposes, two aspects need to be completed. One is the establishment of models about USM's output qualities (the efficiency, torque, power, etc. ) and control variables (the voltage amplitude, frequency, and phase difference). The other one is the research on the multi-variable and multi-objective optimal control algorithm suitable for USM. In general, deep research on USM's three control variables and their relationships, comprehensively making use of the advantages of different control strategies, and implementing the multi variable optimal control methods composes essential contents of research on and development of servo control techniques for ultrasonic motors. In the chapter, a traveling wave rotary ultrasonic motor is used as an example to illustrate the control techniques.
13. 1
Classification of Control for Ultrasonic Motors
According to the structure, principle, and practical requirements of ultrasonic motors, control techniques for USMs motor arc divided into two levels: CD overcome the inherent imperfection of USM, make it operate stably, and obtain high quality output; @on basis of the stable and reliable operating of USM, investigate servo control techniques toward the engineering applications to implement precise position or speed eontrol- 2-. 1. The stability control of USM USM's operating is based on the vibration of the stator excited by the piezoclee-
C. Zhao, Ultrasonic Motors © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011
386
Ultrasonic Motors Technologies and Ap plicalions
tric ceramic element combined with the friction between the stator and rotor. The temperature of its body will increase with cxtcnsibility of the opcrating timc. Thus, thc stator's natural frequcncy will vary, which will cause the variety of the operating condition and output performance. In addition, some phenomena such as abrasion, vibration, and noise will occur during operating too. To dccrcasc or eliminate thosc impcrfcctions is thc highlight of carly stability control for the ultrasonic motor. ='Jowadays, with intcnsive research on USM and increasing acceptancc of its products on market, furthcr improvemcnts of USM's pcrformancc, such as output torque, power, and efficiency, have become important aspects in research on its control techniques.
2. The servo control of USM Since the early 1990s, Yuji, Senjyu, and Fazheng Lin have started with research on servo positioning and speed control using USM. Thcy focuscd on the following subjects: using thc statc, position, spced, and other information of USM obtained by sensors as feedback signals, developing various control algorithms, and using thc amplitude, frequcncy, phase differcncc of a driving voltagc, or thcir combinations as control varieties, to achieve adaptive, high accurate, rapid responsive, and reliable position/ speed control under variable loads applying to ultrasonic motors. Several kinds of control strategics arc coneludcd as following:' 4: : (1) PID controller with fixed/variable gains A PID control system with fixed gains is simple and easy to be implemented, but it is difficult to set the parameters, also, it is not easy to meet the performancc rcquircment of ultrasonic motors. In addition, thc PID control systcm with variable gains, which can overcome the shortcomings mentioned above, but require dctermining relatcd parametcrs according to USM's opcrating statc, cnvironment and load conditions, thcreforc it is not easy to implemcnt in practical applications. (2) Adaptive controllerLs -9An adaptive control is a relatively mature method used in the controlled system with ultrasonic motors, ineluding model reference adaptive controller and selfcorrccting controller. It dctermincs control parametcrs on-line, compares output with referencc model, and compcnsatcs displacemcnt/speed crror in time. (3) Neural nctwork controllcr: 10 10J ='Jeural network control obtains output and input signals' nonlinear mapping function using neurons and connection weights, and is adaptive, self-learning, and fault tolerant. Combincd with specialty of ultrasonic motors, rescarchcrs such as Fazhcng Lin, havc widely invcstigated the application of the neural network in thc control systcm with ultrasonic motors. Gcnerally, thc ncural network controllcr can obtain bettcr control accuracy. However, with the growth of the network's complexity, there will be large number of experimental data to train the network, which will slow the calculation speed and the learning procedure. As a rcsult, thcse factors may limit the rcsponsc specd of the control sys-
Chapter 13
Control Techniques for Ultrasonic Motors
387
tem. (1) Fuzzy controller1l1 - 15 A fuzzy control is an intelligent control method using theory of fuzzy-set and fuzzy-reasoning. This method docs not need the mathematical model of USM. and only needs to make up fuzzy inference rule and choose fuzzy reasoning method according to the operational experience and data. The fuzzy controller is adaptive and robustness. however. under certain conditions that system parameters arc variable. its motion locus will be variable dramatically. A high accurate servo control will be difficult to be implemented. Therefore. the fuzzy control is usually implemented together with neural network control in engineering. (5) Mixed controller: 16 - 19 ]
Because control strategies mentioned above have their advantages and disadvantages themselves. thus researchers developed many mixed types of controllcrs. such as a fuzzy neural network. fuzzy PID. adapting fuzzy. sliding variable structure model tracking controllers. etc., which can be mostly applied to the control system with ultrasonic motors. and can make the control precision with some improvements. but still cannot be used for onsite engineering application. Recently. techniques such as the nonlinear control theory and the genetic algorithm are introduced into control techniques using ultrasonic motors. and the control effect and prospective for engineering applications arc still need to be improved. In conelusion. with USM's applications to control systems. higher requirements arc put forward in its stability. positioning accuracy. response speed. and so on. High quality servo controllers for USM will be developed. as that for electromagnetic type servomotor or stepping motor.
13.2
Speed Adjusting Mechanism and Control Methods of USM
According to the operating mechanism of USMs. the essence of control lies in the adjustment of the vibration amplitude and elliptical trace of points of a stator. and the operating frequency of USM. Therefore. control parameters should be the amp Ii tude. frequency. and phase difference of voltages-3-1, 20_
13.2. 1
Voltage Amplitude Adjusting
According to the forming mechanism of a traveling wave in a piezoelectric vibrator and an elliptical motion equation. adjusting the amplitudes of two standing waves can change the amplitude and shape of traveling waves. There are two possibilities: equal vibration amplitude between two phases and unequal vibration amplitude between two phases. According to Chap. 5. when amplitudes of two standing waves arc equal and their phase difference is 90°. the tangential motion speed of points on a stator is
V"
=-
IT Wn Wo
~
sin( 2;..IT,T -
wnt )
(13. 1)
Ultrasonic Motors Technologies and Ap plicalions
388
Within certain ranges, the amplitude of the standing wave Wo is proportional to the driving voltage amplitude. Therefore. a linear speed adjusting can be obtained through adjusting the exciting voltage amplitude imposed to a piezoelectric ceramic element. Fig. 13. 1 is the diagram of the relationship between USM's speed and voltage amplitude. Due to factors such as the performance of piezoelectric material, the friction between the stator and rotor, nonlinearities, and so on, this diagram displays a threshold of the exciting voltage. which is piecewise linear with rotary speed. 40.16kl-lz 4032kHz
120 :=
';: -S
100
1l
SO
'i3 ~
1:0
'0
e.:
40,49kHz 40_65kHz 40.S2kHz 40,98kHz
4 1.1 5kHz
60 40 20 0
100
80
Fig. 13. 1
120 VoltageN
140
USM's speed vs. voltage amplitude
However, when amplitudes of two standing waves arc not equal and with 90 phase difference. the tangential motion speed of the stator is
V"
=-
IThw;
W + W~;ti\ B [
1
(1
+ eos2wnt) ]sin (2A
1(z: -
0
wnt) (13. 2)
Where. W A and W B represent the amplitude of two standing waves, respectively. Therefore. modifying one of the amplitudes of two standing waves can also change speed. However, non-equality between two amplitudes of standing waves will result in the distortion of the elliptic traj eetory of points on the surface of the stator. Consequently. the points on the stator will contact with the rotor unevenly, and the speed of the USM will be unstable. Thus, this speed adjusting method is strongly nonlinear and rarely used in practice.
13. 2. 2
Frequency Adjusting
If the propagation speed of the traveling wave in the stator is a constant
Cf,
then
Afe' where fe is a exciting frequency. When amplitudes of the standing waves arc equal and their phase difference is 90 according to Eq. (13. 1), the tangential speed of the partieles on the surface of the stators is
Cr =
0
V"
=
,
h. (2IT 2IT 2 j, Wo -sIn -_t-,x - 2IT j-,t ) Q
Cf
Cr
(13. 3)
Control Techniques for Ultrasonic Motors
Chapter 13
389
According to the above equation, adjusting the exciting frequency can control the resonance state of the stator, and regulate the speed of ultrasonic motors in sequence. However, for this kind of control method, there is non-linear relationship between the speed and frequency, as shown in Fig. 13. 2. From the figure, it can be concluded that under small ranges of the frequency variety. the relation between the frequency and the speed can be considered as a linear one. 200 180
.:: 160
-
-
-
-
j
· ,
.
-
-
-
-
-
~
.o
140
.o • • .o •
.o~ • • • .o •
-g 120 0>
.o . . . .o •
.o
"
"S
>:'
.'"
0.
~
<5
0<:
100 80 60 40 20
I
-
-
-
-
-
-
,.
· •
, ,
.: . .
-
.
. '- -
.o ~ •
.o • .o
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.o .o •
.o •
.o.
-
-
-
-
-
.
~ - •
..
•
,,
I
•
'0• • • • • • •
.
.
~.o
-
-
-
-
_
.. .
I
,
•
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-
-
-
-
I
.. .o. •• .o • • • •
.....
.
.o • •
~
.
.o.o
.;. •
.o .o •
•
.o •
~.o • • .o •
.o
.
.: • • .o • • • • ..
' "
-
• • • .o • •
.,
......... ,
~
,•
. . . . .o •
.o
I
, .., , .,.. .o. ..... •• .,, , I.............. \ ................. . . . . . .' ....... . ...... · ............. . . .............. . ., ..... . . , , . . ........ . ., • • • • • .o
·· ·
-
... ... .o. -:••
~
I
.o • • .o ..
,.w .. . .
-
.o • •
I
, , , ,
.o • • .o •
•
~.&..
.
~ - - -
•
I
, ,
•
.o . . . .o •
.o
~
.o
• ,
. . . . . . . . . . .o.o • • •' • • • • • • • ,
I ,
•
• • • • .o •
,
.o • • .o
• •
,
.. .o •
•
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~
• •
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•
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I
•
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••••• •
Fig. 13. 2
• •
I
.o • • • • • .o • • .o •
•
I
40.5
.o
I
..
\.o ...
.o •
.o
. o . o .
• • •
\ • • • • • .o
I I
•
0
~.-
•
~
. . . . . . .o
0 40
13. 2. 3
. ·· · .o:·• ~ -
-
ow . . . . . . . . . . . . . . . . . .
.o . . . . .o •
. o . o .
I
•
~
•
~
41
41.5 42 42.5 Dri ving freqllencylkHz
I
~
43
43.5
USM's speed vs. driving frequency
Phase Difference Adjusting
When the phase difference between the standing waves in the stator is a random value a rather than 90°, the traveling wave combined can be obtained according to Chap. 5 w
=
W 0 sIn . T-T 2rc. slnwnt
+W
0
2rc. cos T-T sIn ( wnt
+) a
(13. 1)
It can be inferred that the speed amplitude of the tangential motion of the traveling wave's crest is rcwnh Wo sina I V" I = ~;=::::;;::===::::c::::::;===
A
,J sin' wnt + sin' (wnt
+ a)
(13. 5)
From the equation above, it ean be eoncluded that the speed of points on the stator is a function of the phase difference between standing waves. Therefore, when the exciting voltage and frequency are fixed, adjusting the phase difference between voltages can also change USM's speed. This method also has the problem that the phase difference is not nicely linear with the speed. Fig. 13. 3 shows the diagram of the relationship between USM's speed and phase difference when USM is in practical use. When the phase difference varies between - 90° and 90°. the speed of USM varies from maximum clockwise speed to maximum counterclockwise speed. However, there is a dead zone between - 20° and 20°, and the relation between the phase difference and the speed is nonlinear. Table 13. 1 is the eomparison of these three control methods. Results show
Ultrasonic Motors Technologies and Ap plicalions
390
150 100 50
~
--E
>::'
"'"
0
C
- 50
~
e0
cr::
- 100 _1 50 L--L__~~__~__L--L__~~__~~ -1 00 - 80 - 60 -40 - 20 0 20 40 60 80 100 Phrase diffcrcnce/(")
Fig. 13.3
USM's speed vs. phase difference
that the frequeney adjusting is relatively suitable for speed eontrol, while the phase difference adjusting is relatively suitable for the position control. Table 13. 1
Comparison of spced adjusting mechanisms for ultrasonic motors
Control
Adjusting
variables
mechanism
Amplitude
Frequency
Phase
13. 3 13. 3. 1
Advantages
Change wave ampli-
Linear speed adj usting
tude
Simple controller structure
Change resonant state of stator
Change elliptical trace of point
Quick response Low speed start up Simple power supply
Disadvantages Small adj usting range Large dead zone Small torque and low speed
Nonlinearity Instability
Easy and smooth reversing
Difficult to low speed
Steady speed adjusting
start up
Easy to control
Complicated electric circuit
Stability Control Techniques for Ultrasonic Motor Principle of Frequency Automatic Tracking- 202lj
The stability of USM is mainly influenced by temperature. This section introduces a frequency automatic tracking (FAT) method used for USM's stability control. A piezoelectric ceramic element is sensitive to temperature. After operating for a long time, the temperature of USM will increase, as shown in Fig. 13.4, and the resonance frequency of a stator will decrease, then the speed of USM drops under open loop condition, as shown in Fig. 13. 5. In this figure, 51 is the speed vs. frequency curve of USM before the temperature increases. 52 is the speed vs. frequency curve after the temperature increases. Moreover WI' and w'p are corresponding resonance frequencies of the stator, respectively.
Chapter 13
Control Techniques for Ultrasonic Motors
43n=---------------------,
391
11
~ 42.5
,.,
~
~
42
~
4 1.5
ff
g
II ,
"81"/.1 5i' C
'"
'0
~
<>
41
<>
<>
Angular rrequency
Temperalurel "C
Fig. 13. 4 Stator's resonance frequency vs. temperature
Fig. 13.5 USM's speed vs. frequency under different temperature
Supposing that P j is the operating point of USM which corresponds to the rotary speed n1' before the temperature increases, when the temperature increases, and the characteristic curve of the ultrasonic motor changes from 51 to 5,.
If the driving signal frequcncy is fixcd, thc operating point will be P" rotary spced of USM will decrcase to
n,.
and thc Thus, thc drivcr decreases automati-
cally thc driving signal frcquency, which makcs the operating point to movc to
P3
,
so as to keep the original rotary speed nj. This is frequency automatic track-
ing mcthod. When thc rotor of the ultrasonic motor only contacts with thc wave crcst of thc points of the stator surface, according to Eq. (13. 1), thc rotary spccd nand thc axial amplitudc of thc stator's bending vibration Wo havc thc following rclationship (13. 6) where a is a proportional constant. Therefore. the rotary speed of the motor is proportional to the stator's amplitude and the exciting frequency. Actually, the amplitude is influenced by many parameters, the most important one is the exciting frequency (13. 6) can be rewritten as n
=
awo W Ot Cwo) (
~
)
Wo'
Thus, the Eq.
(13. 7)
In Eq. (13.7). thcre is a t added to the suffix of the amplitude. which means that thc amplitudc is time variablc. That is to say, cvcn thc frequency is constant, the amplitude still changcs slowly. If therc is a minor variation L thc operating point
Wo ,
w
ncar
then
The reason that this formula can be approximated in calculation is Lw/ Wo
< 1%
gcnerally. According to Eq. (13. 8), if the exciting frequency can be adjusted
392
Ultrasonic Motors Technologies and Ap plicalions
+
online to make W Ot (wo 6.w) = const, we can guarantee the stable rotary speed of the rotor. This is the principle of the frequency automatic tracking method.
13. 3. 2
Detection of Amplitude
In order to make the rotary speed of the rotor stably, the vibration amplitude of the stator must be constant. Then, how to detect the vibration amplitude of the stator? Generally a method used is to install a piezoelectric ceramic transducer in the piezoelectric ceramic piece pasted on the stator. This transducer is the isolated electrode mentioned in Chap. 5. Under the action of the traveling wave, the isolated electrode will generate alternating voltage induced by the piezoelectric effect. According to Eq. (2. 16b) , and letting 1=0, the voltage is . V I = - : - - -K C WI
lw
(13. 9)
./0
where Co , K , and WI represent the elamped capacitor of the isolated electrode, the force coefficient of the piezoelectric ceramic piece, and the vibration speed of the isolated electrode, respectively. Since W Ot is a slowly variable signal WI
=
tt (Wotsinwt) """ wWOteoswt
(13. 10)
Substituting Eq. (13. 10) into Eq. (13. 9), we can obtain:
VI
= -
KWOt
--'------c coswt 1
(13. 11)
~o
Therefore, the alternating voltage of the isolated electrode is an alternating signal having the same frequency as the exciting frequency, its amplitude is proportional to the amplitude of the traveling wave in the stator, and the rotary speed is also proportional to the amplitude of the traveling wave in the stator. Therefore, it can be conel uded theoretically that if the transmission between the stator and rotor is ideal, the amplitude of the alternating voltage on the isolated electrode is proportional to the rotary speed of the motor. If the alternating voltage VI of the isolated electrode is rectified and filtered furthermore, the average voltage obtained is also proportional to the rotary speed of the motor.
13. 3. 3
Implementation of FAT System
Figure 13. 6 shows a elosed loop control system FAT. In the system, the feedback voltage comes from the isolated electrode, which is also called as an isolated voltage. Experimental result shows that the load not only influences the rotary speed, but also influences the isolated voltage. That is to say, under the same rotary speed, if the load is different, the isolated voltage is also different. There is still no reasonable explanation yet at present. Because the operating region of USM is often between the resonance and antiresonance points, therefore the operating frequency locates in the right side of the maximum rotary speed. In Fig. 13. 6, the isolated voltage is applied to the in-
Chapter 13
Control Techniques for Ultrasonic Motors
393
Given voltage ,...-------,
+
Block diagram of the feedback electrode voltage feedback control system
Fig. 13. 6
put terminal of PI controller. When the whole system is stable, the isolated voltage equals to the given voltage. The speed adjusting of USM can be obtained through changing the given voltage. If the given voltage is a constant, the USM's speed is also constant under constant load. Figure 13. 7 shows the relationship between time and the rotary speed under open loop and the temperature. At the same time, Fig. 13. 7 also shows the relationship between time, the rotary speed, and the frequency under elosed loop of the motor. From this figure, after the frequency automatic tracking control (closed loop) is used, the variety of the motor's rotary speed is kept within 5 %. As the temperature of the motor increases, its open loop speed drops dramatically. Under constant load, the frequency automatic tracking technique can fairly compensate the variation of the motor's speed caused by temperature change. 150
55
140 '-
~
~
""s;-
50 !-l
130
~
120
J:
45 [
'-,
~
~
Il O
C ~ 0 100
""
N
B
40
90 35
80 70 60
30 0
5
10
15 {/min
20
25
30
25
Fig. 13.7 Experimental result of rotary speed varies with temperature in TRUM-60
13. 4 13.4.1
Ultrasonic Motors Used as Servo Motors Ideal Servo Actuator-USM
As equipment to transform the electric energy into mechanical energy, as viewed from applications, an elcctromotor can be divided into two kinds: a power motor and servo motor. They have the same operating principle, but the different function. The significance of the power motor lies in exporting a large enough power
394
Ultrasonic Motors Technologies and Ap plicalions
and generating a large enough torque (or force), so as to drive mechanical facilities. The usage of the servo motor lies in that it can change its operating state, speed, output torgue (or force), etc., to adapt quickly to the continuously variational operating condition. In addition, their operating state, performance, and requirements are also different. In general, the servo motor is used in the electromechanical servo system as an actuator, whereas the power motor is used for general mechanical motive power systems. According to the operating demand of servo systems, the following requirements arc put forward to the servo actuator: (1) The output torque and rotary speed can meet the requirement of load. (2) A relatively wide adjustable range of speed and torque, and controllability in low speed operating. (3) A rapid response, that is to say, a quick start and stop, or reverse, and a quick response to signal variation. (1) A large torque/mass ratio and small volume as far as possible. (5) A linear relationships between the rotary speed and torque, and between the torque and control variable. According to above requirements, and compared with the characteristics of an ultrasonic motor, it can be seen that the performance of the ultrasonic motor are good enough to make it an ideal and small servo motor. However, as a novel servo actuator, whether or not the ultrasonic motor can be widely used for the servo system and whether or not its excellent performance can be given are determined in a large extent by whether or not it is convenient to be controlled and whether or not it has good control performance. Therefore, profound and systematical research should be carried out on the servo control techniques using ultrasonic motors.
13.4.2
Requirements of Servo Control Using USM
The servo control using ultrasonic motors mainly means its position and speed controls, and sometimes also includes its output torque control. However, whether the position or the speed controls, the servo control system based on ultrasonic motors generally includes following components: transducer used to feedback the position or speed of USM; an ultrasonic motor used as an actuator to implement the motion, such as the mechanical objects driven by USM; the controller used to implement a control strategy and a control algorithm; a servo driver used to transform and amplify the control signal exported by a controller to make it meet the requirements of the format, energy, amplitude, power of driving, etc. In the design of the servo system based on ultrasonic motors, we must consider that: (1) The stability of operating, which is the foundation to make sure the position and speed of a motor to track a command. (2) The accuracy of tracking, which means the output position or speed of a system is very close to a target.
Chapter 13
Control Techniques for Ultrasonic Motors
395
(3) The rapidity of response, which means the system arrives at a target value within a time as short as possible. (4) The robustness of control, which means that under situations when the characteristics of USM change or there is external interference, the control performance of a system can be kept. Many strategies and plans have been applied to the accurate servo control of ultrasonic motors. Limited by the space, this chapter will only introduce several usual and already applied control strategies. For other control strategies, please refer to relative writings.
13. 4. 3
Servo Control System Using USM '
1. Control system based on computer The implementation of control strategies IS based on reliable detection and control equipment. For example, the speed control of ultrasonic motors need to detect precisely the instantaneous rotary speed of USMs, while the position control needs to measure precisely the current position of USMs. For this sake. in order to obtain a superior man-machine interface to implement different control strategies, the PDLab made the robot arm, which is based on ultrasonic motors, as shown in Fig. 13. 8. This system ineludes: three traveling wave rotary ultrasonic motors. each of which cones ponds to a driving and control circuits, a rotary photoelectric encoder, a frequency/voltage transformation circuit. a motion controller, and a computer used for all USMs, as shown in Fig. 13. 9. Because the following control testing completed are all based on the system, we can compare with the control quality of various control strategies.
Fig. 13. 8
Robot arm based on ultrasonic motors
In Fig. 13. 9. when the motor rotates, the rotary photoelectric encoder gives electric pulses. which can indicate the position of the motor. When the rotary speed of the motor varies, the frequency of the electric pulse from the encoder varies accordingly. The computer determines the instantaneous rotary speed of the motor according to the voltage sampled by the frequency/voltage transformation circuit. as shown in Fig. 13. 10. simultaneously obtains the position of the motor by reading the values from the counter, then determines the corresponding
396
I
Ultrasonic Motors Technologies and Ap plicalions
J--f
USM
-;..:..::.:..:.---------
State
~
I
Driving and control circuit
Rotary photoelectric encoder
Mechanical load
I
" " ~~
--- --- -- -- --
~o
BlO
r
RlS CW/CCW
§
'" u
'-------
I
.. _-------Servo control signal
Fig. 13. 9
DAC
I
Timer interrupt request
] '8 U
ADC
l-
I
I I
-----t--------+--------+-----------Computer system (control algorithm) I
~
I I
~
'13
""'"
Ii
I I I I
IReversible counterl
~ "
I I I
!
~S
r-
GO~400b;;d-------------i
IQuad frequency processing circuit I
> "
.~
Frequency/voltage converter
-" "'" "'"
u
~;§
~~
T
--------_ ..
I
Composition of control system based on computer using ultrasonic motor
controlled variable using certain control algorithm, and finally sends out the control signal to driving and control circuit through the interface circuit. We use the voltage from the isolated electrode on the stator to control the rotary speed of USM. However, no matter which speed detection method is used, the feedback signal needs an AID transformation before they arc sent to the computer.
l'i ..
10
v'"' 8
c,
<:.,
6
~ "0 > 4 2
OR-__ t 11----L__J--"-.......--1__J-'
(a) FN circu it
Fig. 13. 10
o
~
____
~
____- L_ _
~
3 000
6000 9000 12000 1 1Hz (b) Relationship ofFN transfonnation
Relationship of F/V transformation based on a rotary encoder
2. Flowchart of the control program Figure 13. 11 is the flowchart of the control program. In this program. timer interrupt is used to improve the real-time response of the control system rather than timing software. 3. The determination of the system's sampling period The sampling period is the time interval during which the computer scans the change of parameters. calculates and exports the control signal. Choosing an appropriate sampling period is important to guarantee the ultrasonic motor's control
Chapter 13
397
Control Techniques for Ultrasonic Motors
Start
N
Timer interrupt I
N
~ ____L_-_-_-___-_-_-_-_-_T_-_-_-_...J__s:~:c: ~~u~~::
Stop
Fig. 13. 11
Return from interrupt
Flowchart of the servo control program using a ultrasonic motor
quality. Theoretically, the shorter the sampling period is, the higher the control quality is. However, short sampling period will increase the burden of the system's hardware. Especially, when the response speed of the controlled object is relatively low, too short sampling period cannot improve the dynamic characteristic of the system. While setting the sampling period of the servo control system using USM, the following aspects should be considered: (1) The response time of USM. Since USM has the response time of milliseconds (for example, TRUM 60's response time is only lms) , the sampling period should be the level of millisecond too. (2) The capability of the hardware system. It mainly indicates the execution speed of CPU, the conversion speed of the A/D and D/ A converters, etc. The sampling period of the system should be far below their speed. (3) The calculation amount of the control algorithm used. For those algorithms having large amount of calculation (for example, the neural network algorithm), the sampling period should be increased, and on the contrary relatively short sampling period can be used. After the above factors are considered sufficiently, the sampling period is usually set around 4-l0ms.
13. 4. 4
PID Controller Using USM
PID controller has the characteristics of simple algorithm, high preclSlOn and high reliability, etc. It is widely used in the control of industrial objects and industrial processes. PID control includes two types: fixed gain and variable gain. Relatively, the former is easier to be implemented, while the latter can ob-
398
Ultrasonic Motors Technologies and Ap plications
tain a comprehensive and higher control quality. The former is the basis of the lattcr. Thcrefore, the fixcd gain PID control of thc ultrasonic motor will be introduced first.
1. The fixed gain P ID control of USM L3J Figurc 13. 12 shows thc block diagram of the PID control systcm for USM. In this figurc, the PID controller compares thc objcctivc position Yd (t) (or thc speed) of the motor and the actual position (or the speed) yet) , and obtains the deviation signal e(t) = Yd (t) - y(t), which combines linearly the proportion of thc dcviation, the intcgration. and thc diffcrentiation of the dcviation to timc, to compose thc controlled variable. The control law of the PID controller can be cxpressed as U(s)
=
Kp (1
+ 1~ S + TDs )E(s)
(13. 12)
r
wherc Kp is the proportion coefficient. T r , Tn and s arc thc integration, thc differentiation time constants, and Laplace variable respectively.
Proport ion (Kp)
USM
lntegralion ( 1I"Ii.l,)
y(/)
DifTerelll iali on (1;, s)
Fig. 13. 12
Fixed gain PID control system using ultrasonic motors
Whcn PID control is implcmented in the computcr control systcm, Eq. (13.12) should bc discrcte to thc digital form, whosc algorithm is
+ KI b k
u(k) =
Kpe(k)
e(j)
+ KD[e(k)
-
e(k -
1) ]
(13. 13)
j=O
~ h Were Kr = Kp Tr called an integration coefficient, Kn
=
~
Kp T, called a differ-
cntiation cocfficicnt, T, is a sampling period. A key in the PID controller design is the reasonably tuning of the coefficients K p , K r , and Kn. Gencrally, thcre are two kinds of tuning mcthods. Onc is thc theoretical calculation method. Another is the tuning on-line method. which relies on engineering experiences to tune directly the coefficients in the control system established already. Because USM does not have any practical, effective, and control-oriented theoretical model, the only method is to select PID coefficients by trying on-line relied on experiences. Experiments reveal that during the sclection, we'd better sct thc cocfficients from following sequcncc: first a proportion, then a integration, and finally a diffcrentiation. Especially, PID controller using thc ultrasonic motor somctimes only includcs the proportion, or thc pro-
Chapter 13
399
Control Techniques for Ultrasonic Motors
portion plus the integration. I OOr--~---r--~--'---'
- -- --
Kp~ IO
-- K,.~20
0.3
0.4
0.4
0.3
0.5
~
0.5
~
(a) P cOII "ol
Fig. 13. 13
(b) PI cont rol
Responses of P control and PI
control for the ultrasonic motor without load
Figure 13.13 shows the results of PID control without load. Fig. 13. 13(a) is a step response with only P control. Here, the rotor arrives at the target position quickly. However, oseillation occurs near the target position. In addition. Kp influences the rising time of the system, which will decreases while Kp increases, but the overshoot also increases. Fig. 13. 13(b) is the step response after I control is added, which not only the oscillation and overshoot of the motor are restricted dramatically. the control accuracy of the motor is also increased. When the PI control of the ultrasonic motor is implemented. in order to prevent the integration "saturation", Eq. (13. 13) needs a modification. Only when the position error is relatively small, the integration component operates. Figure 13. 14 is the result of PI control with load. Compared with Fig. 13. 13 (b) , without change in the coefficients of the controller. the amplitude of the error increases when the load added. Fig. 13. 15 is the response of PI control when the command is 15°-90° position of the square wave. Thus, it can be seen that PI control can implement preferably the position tracking of the ultrasonic motor. I OOr--~--~--~--r---,
90 r-1
.
80
"§:>
70
:r-!
:r--
r-"I
,,..-,
II :- -- -Expected valuel - Actual value
"::"
"
0;
.2 60 ;;; 0
"" tis
0.3
0.4
Fig. 13. 14 Response of PI control with 10ad(Kp = 20, K, = 2)
0.5
50 1.-
40
0
0. 1
,'-"
....... 0.)
0.2
,'--'
:t.-.
0.4
0.;
I/ S
Fig. 13. 15 Square wave tracking response(Kp =20, K,=2)
In the control experiments, through regulating PI coefficients continuously,
Ultrasonic Motors Technologies and Ap plicalions
400
the effect of the control can be improved. However, because of the randomness of trying cocfficicnts, and espccially bccausc of the strong timc-variation and nonlinearity of USM, which are influenced by the increase of temperature, the interference of load, and also other factors, the parameters of the motor and its speed characteristic both vary. It is hard to select the ideal coefficients for the controller. If fixed gain is still used for PID coefficients to control the motor, the control quality of the system will be reduced. The results of PID control using the motor with a load are enough to prove this point. Under this condition, if we can implemcnt thc advantagc of PID control and obtain good control effect, it is bcttcr to dynamically tune thc gain of PID controllcr during thc opcrating of thc motor according to its operating statc, to compcnsatc the influence brought by the time-variation and nonlinearity of the motor. On-line PID coefficients tuning can be obtained through the fuzzy technique, the neural network technique, the gcnetic algorithm, etc. )Jext, the neural nctwork tcchnique will be introduccd to implement the on-line tuning of PID coefficients.
2. Neural network variable gain P ID control using U5Mr 4 , 22
23-
The gain of PID controller can cmbody the charactcristic of a control system ovcrall. Howcvcr, its robustncss and sclf-adapting are relatively poor. Thc ncutral network has relatively strong ability of approximation in the nonlinearity, self-adapting, and robustness. If thesc two tcchniques arc combincd, which means the coefficients of PID control are tuned on-line through the neural network, namely, whcn the motor is undcr differcnt operating conditions or its inner paramctcrs vary, and diffcrcnt coefficients of PID are selected on-linc to control the objcct, theoretically this method will obtain better control quality than thc fixcd gain PID control. In addition, this technique will omit the complicated proccss in tuning the coefficicnts of PID control. (1) Basic ideology of the ncural nctwork PID control In order to explain the essence of the neural network PID control, Eq. (13. 13) is modified as an increment equation Lu(k)
=
Kp[e(k) - e(k - 1) ]
+ KD[e(k) -
+ Kre(k) + e(k -
2e(k - 1)
(13. 11)
2) ]
Eq. (13. 14) is diffcrcnt from the fixcd gain PID. In the variablc gain PID control algorithm, the values of K p , K r , and Kn depend on the adjustable parameters of thc operating condition of thc system. Thus, abovc equation can bc writtcn as more general form Lu(k)
=
r3(K p ,Kr ,Kn ,e(k) ,e(k -1) ,u(k -1) ,e(k - 2»
Where r3(') is the nonlinear function relating to K p
,
K"
KD
,
(13.15)
e(k), e(k -
1) ,
u(k - 1), and e(k - 2). Lots of methods can be used to approximate r3('). If BP
(Back-Propagation) nctwork is uscd to approximatc r3( • ), the neural nctwork PID control is made up. (2) Structure of the neural nctwork PID control using ultrasonic motor Thc neural nctwork PID controller using the ultrasonic motor is shown In
Chaptcr 13
Control Tcchniqucs for Ultrasonic Motors
401
Fig. 13. 16. It includes two parts: CDThe PID controller controls directly the objcct, whosc cocfficicnts arc variablc; CZ)BP ncural nctwork adjusts on-linc thc cocfficicnts of PID controllcr according to thc opcrating condition of thc systcm to make certain performances of the system best. The output of this neural network corresponds to the three adjustable coefficients of PID controller.
Fig. 13. 16
:'\feural network PID control system using ultrasonic motor
Figure 13. 17 is the topological structure of the neural network used, which has one hidden layer. Theoretically, BPNN (Back-Propagation Neural Network) with onc hiddcn laycr can approximatc to any nonlincar function, and with thc incrcasc of thc nodcs in thc hiddcn laycr, thc approximatc accuracy of thc nctwork increases too. But the calculation amount of the network increases dramatically. Therefore, the nodes of the hidden layer are set to six. According to Eq. (13.15), thc input of thc nctwork is sct as e(k), e(k-l), and e(k - 2). Thc cxcitation function of thc ncuron in thc hiddcn laycr is dctcrmincd according to the following equation: tanh(.r)
(13.16)
e(k) e(k-l) e(k-2)
].F3
Fig. 13. 17
{F6
Structure of BPNN network used for the control of object
Because the coefficients of PID cannot be negative, the excitation function of thc ncuron in thc output laycr uscs thc nonncgativc Sigmoid function
Ultrasonic Motors Technologies and Ap plicalions
402
[l+tanh(.:r:)] 2
17;,(.:r:) =
.
(13. 17)
In order to avoid saturation during the operating of the neural network, the input of BPNN can be normalized. Because the value of the output layer excitation function g, (x) is within (0,1) , the output of the neural network needs to multiply an appropriate proportional factor to become the coefficients of PID controller. (3) Forward calculation of BPNN The forward calculation is a process for BP:'\JN to determine the values of Kp , K J , and K D • From Fig. 13. 17, for the nodes in the input layers of the neural network, their input/ output relationships are (1)
{
OJ
_
-
ajY
e(k - j),
.Tk-j
=
I,
M= 3
j=O,1,2
(13. 18)
For the nodes in the hidden layer, their input/ output relationships are
b 3
net;2) (k) {
0;2)
=
(k) =
o~) (k)
aji) (k)
f[:~t;2) (k) ] , 1,
Q
i
=
0,1, ... ,Q - 1
(13.19)
6
=
where is the connection weight between the node j from the input layer and node i from the hidden layer; w;}} (= 8i ) is the threshold value, and the superscript (1), (2), and (3) represent the input layer, the hidden layer, and the output layer respectively. For those nodes in the output layer, their input/output relationships are 6
{
net Z3 ) (k) a;') (k)
=
=.
~ W~3) a;2) (k)
g, [net;') (k) ] ,
(13. 20) l
=
0,1,2
is the connection weight between the node i in the hidden layer and where the node l in the output layer; wg) is the threshold value; and wg) =8,. Thus, the coefficients of PID controller is
Kp (k)
=
Gpae') (k),
K, (k)
=
G,a;') (k),
KD (k)
=
GDoi') (k) (13. 21)
where G p , G" and G D are the proportion coefficients selected properly. (4) Adjusting of the weight in the BP:'\JN In order to adjust the weight of the network, the modifying function of the weight is set as (13. 22) The steepest descent method is used to modify the weight of the network, i. e. according to the criterion J to search and regulate under the direction of the neg-
Chapter 13
Control Techniques for Ultrasonic Motors
403
ative gradient, and plus an inertia term which makes the search rapid convergent to a global minimum value. Thus, the modified amplitude of the weight is
!:::"w~') (k + 1)
=-
"I
a~3) + a!:::"w~') (k)
(13. 23)
aW/i
where r;( > 0) is a learning rate; a (~ 0) is a smoothing factor; a!:::"w~') is a inertia term. While a]
a] ay(k
ay(k+1) duCk)
+ 1)
aw~3)
duCk) ao;3)(k)
ao;3)(k)
anet;3) (k)
anet;3) (k)
aw~3)
(13. 24) According to Eq. (13. 11), following equations can be obtained
J
du (k) (3) (
ao o
l
k
=
)
Gp [e(k) - e(k - 1) ]
du (k) , aO ;3) (k) = Gle(k) duCk) (3) (
ao,
k
=
)
(13. 25)
GD [e(k) - 2e(k - 1)
+ e(k -
2) ]
Therefore, the adjusting amounts of the connection weight in the output layer of BP='JN are !:::"W~3) (k
+ 1) =
r;B;3)
0;') (k) + a!:::"w~3) (k)
J B(3) (k + 1). ay (k + 1). duCk) • (k)] e duCk) Cb;') (k) g, ne I g: [net;') (k) ] g, [net;') (k) ] {I - g, [net;3) (k) ] } , l l=0,1,2 =
I
t (3 )
[
I
I
(13.26)
=
Similarly, according to the above predication method, the adjusting amounts of the connection weight in the hidden layer are
1
!:::,.w~') (k+1)= r;B;')o;]) (k)+a!:::"wt2)
B;2) =
j' [net;2) (k) ]
f' [net;2)
(k) ]
=
1-
~ [B;3)w/i (k) J, i
0,1, ... , Q-1 (13.27)
=
f [~et;2) (k) ]
The modification of the weight needs to calculate the partial derivative term
dy;~(t) 1) ,
which involves the model of USM. However, currently there is no
effective model of USM. In order to resolve the problem, generally, the symbol-
. f· [d y (k+1)]. db· dy(k+1) lC unctIOn sgn duCk) IS use to su stltute duCk) . Finally, the calculation formula of the weight value in the hidden layer and the output layer are coneluded, respectively: (k
+ 1) =
(k+ 1) =
(k)+!:::"w~')(k+1), l=0,1,2,3
(k)
+ !:::"wt2) (k + 1) ,
i
=
0,2, ... , Q
(13. 28)
Ultrasonic Motors Technologies and Ap plicalions
404
(5) Control results Figure 13. IS shows experimental results of the neural network PID control using USM. In Fig. 13. IS (a), the neural network PID control not only obtains relatively high position control accuracy (stable state position error: -0. oS~+ o. OSO). but also keeps original control quality of the system even when the load is added. Fig. 13. IS (b) shows that USM possesses rapid and high accuracy servo position control performance. which modify PID controller's parameters online through neural network.
.
.
_ . 71.-0.0 N· III -
>::'
7i.- O.2 N ·III
>::'
'"
90
80
a
0;,
'"
60
«'"
"2
'"
40
0;,
a
.2 0;
0;
« 0 0
0.5
2
1.5
- - Aci llal val lie
. - - - -Expecled val ue
20 0
0
2
4
I/S
(a)
.,t
40
5 E
30
'"0. a 20
a:
10
I\... V'"
It"" ......,
. :
. J\ jL fl 1'- r~
I
10
2 ,------------r-----------,
~ ---~ It
m {l)
" i,
-K
-
'1 ..J
~
OK!)
~
~i·~_(~:L · X=~.--L:L_j~~:C~
~
8
(b)
50
~
6 I/S
~
-
)
i 'f
10
o o
n ,c)
"" 5
lIs
I/S
c)
(d)
10
Fig. 13. 18 Experimental results of the neural network prD controller using USM
13.4.5
Adaptive Controller Using USM L5 • 78J
It is well known that the performance of USM is influenced by temperature, load. pre-pressure, and its driver. etc., and it has strong time-variation and noncertainty. The good operating of PID controller can make a system obtain relatively high control accuracy within short operating time. The parameters of the system vary after operating for a long time, therefore, it is hard on PID control to keep the system in a superior control quality. Under this condition, a selfadapting control will be a good choice. The self-adapting is a control method adjusting the parameters of the controller continuously to compensate the variation in the characteristic of controlled system, and especially suitable to system with time-varying and nonlinearity. The self-adapting control mainly includes a self-
Chapter 13
Control Techniques for Ultrasonic Motors
405
tuning control and the model reference adapting control. The application of these two techniques in the control system using USM will be discussed as follows.
1. Self-tuning control using USMC3· 24 2S] In all kinds of the self-tuning control methods, the self-tuning controller usmg the generalized least square error control is most suitable to the inverse unstable system. Therefore, we introduce this algorithm into the servo position control system using USM. Fig. 13. 19 shows the block diagram of the self-tuning control system using USM. The kernel of the self-tuning control system is to identify dynamically the parameters of the USM through identifier to determine the control function of the controller, to keep the error of the system least.
I
'--""T""---,I-+-Y_(k..> Object
I
Block diagram of self-adapting control system using the ultrasonic motor
Fig. 13. 19
It is supposed that the mathematical model of the motor can be de script as the following difference equations
+
AY(k) = q-dBu(k) w(k) { A = l+alq-l +···+anq-n
B
=
bo +b1 q-l
+ ... +bmq-m
(13. 29)
where d is the delay of the control; q -1 is the reverse shift factor; w(k) is white noise; nand m are the order numbers of A and B, respectively. Since this system is a precise position control system, the controller accepts the expected position and the actual position as inputs, and chooses the phase difference as output. In the self-tuning controller, generally, the principle of the control is to make the following cost function 11 least (13. 30)
where Y and Yd are the actual position and the expected position, respectively; A is the control factor, which is to limit the very large input of the control, and improve the stability of the self-tuning closed loop system. In order to obtain the input which can make the 11 least, an auxiliary system is defined as r(k
+ d)
=
y(k
+ d)
- Yd
+ AU (k)
(13.31)
The corresponding object function is
12
=
E{[r(k+d)]'}
It can be proved that u can make the obj ect function
(13. 32)
12 of the auxiliary system
Ultrasonic Motors Technologies and Ap plicalions
406
least. and can also make the original object function 1] least. The general optimal expectation model of the output for the auxiliary system is rO (k + d/ k)
yO (k + d/ k) - Yd + AU (k)
=
(13. 33)
whereyO(k+d/k) is the optimal expectation of the y(k+d) in time k+d. and composes of time k and the previous information. Let reek + d/ k) = O. and obtain directly the control law of the system u(k)
Yd -
=
yO(k
A
+ d/k)
(13. 31)
Because
=
yO(k+d/k)
(13. 35)
Gy(k) +BFu(k)
where G and F are determined by the equation of Diophantine
1
FA+q-dG
=
1 F. G
1 + flq-l + ... + fnfq-nJ f!;o + f!;l q-l + ... + f!;ngq-ng
= =
degF degG
=
Thus. we obtain the input of the control which makes (k)
Yd (k) - l'Y (k) BF+A
=
u
or
B
1
u(k)
=
ho
f'..
11 least
Yd (k) - l'Y (k) M+A
B
(13. 37a)
=
ng
+ A [Yd (k) -
(13. 36)
d-1 n-1
=
f!;iY (k -
i) -
miu (k -
(13. 37b)
i) ]
Because the parameters of the system a i and hi are unknown. and they are needed to be identified. we define the cost function 1, as N
13
(13.38)
L-pN-k{y(k) -
=
where cf>(k)
[y(k -
=
{ (J =
From
;11;
=
1) .y(k -
2) .... . y(k - n).
u(k - d) • u(k - d [ -
a2 ••••• -
al • -
an
1) •...• u(k -
.ho .h 1
•••• •
d - m)
r
(13. 39)
hmJ T
O. we can obtain the estimation algorithm of the recursive least
mean square for the parameter
J
O(k)
=
O(k -
+
1 ! K(k)
=
P(k)
=
p
(J
1)
+ K(k) [y(k)
- cf>T (k) O(k -
P(k - 1) cf>(k) cfir (k) P(k - 1) cf>(k)
[P(k -
1) - K(k) cf>T (k) P(k -
1) ]
(13. 40) 1) ]
where p is the oblivion factor. whose range is [0.9. 0.99].
Chapter 13
Control Techniques for Ultrasonic Motors
407
Generally, the less the value of p, the stronger the ability of tracking time-varying parameters, and simultaneously the severer is the influence of the noise interference. Because the system with USM is a slow time-varying system, set p as o. 99. The closed loop equation of the system is
_ L!L Yd +;. +BFe(k)
y(k) -
A;. +B
A;. +B
(13.11)
When;' = 0, the closed loop characteristic equation of the system is B = o. If it has any zero pole residing outside of a unit circle, the system is an irreversible stable system, namely the closed loop system is unstable. Thus, selecting a suitable;' can control the reverse stable system. The block diagram of the system is shown in Fig. 13. 20.
Fig. 13. 20
Program block diagram of least square error control
2. Model reference adaptive control using USM L1J The model reference adaptive control (MRAC) is a kind of control method based on model. It adjusts the parameters of the controller to compensate the change of the parameters of the controlled object. This method does not have high requirement of the model of the controlled object. According to the operating characteristics of USM, this kind of method can be considered in the control system using USM. The MRAC position control system using USM is shown in Fig. 13. 21. In the MRAC system, a reference model G m (5) is set, its controller is composed of the pre-filter it, the feedback compensator F (t) and the self-tuning mechanism. ret) is the reference input of the system. When USM operates, the system regulates dynamically the parameters of the controller Ie and F(t) through the generalized error of the position em' which is obtained through comparing the actual position x, (t) of the motor with the output position x'" (t) of the reference model
Ultrasonic Motors Technologies and Ap plicalions
408
G m (s) • to compensate the error introduced by the nonlinearity of the motor and the changes in the parameter of the motor. to make the actual position of the motor close to the reference modcl as much as possible. In the figure. the frequency modulation method is used to control USM which uses the variation of the frequency as the control variable. The value of this variable is added eventually to the reference frequency to form the driving frequency of the motor. In the control methods there arc three important problems:
f*(/)
Reference model
r (t)
x,(/)
USMmodel
'------I Self·adapting 1--_ _-' mechanism
MRAC control system using USM
Fig. 13. 21
(1) An approximate transfer function model of USM Because the speed response characteristic of USM possesses that of the first order inertial element. and the MRAC has low requirement of the model. therefore. the transform function of USM can be supposed as G( ) I
S
=
D,(s) = UI(s)
kUSM res",s+
1
(13. 12)
where D, (s) and U I (s) are the expressions of the motor rotary speed and the controlled variable of the frequency converter in the frequency domain. respectively; rUSM is the time constant of the motor; kUSM is the proportional gain in the model.
For a time-varying object. kus", contains certain amount of time-varying uncertainty. In the design of the MRAC. kus", can be considered as constant. According to Eq. (13.12). the nominal transfer function of the motor uSing the position as output is CPOS
=
(13.13)
s( res", + 1)
where @(s) is the expression of the angular displacement of the rotor in frequency domain. a o = l/rusM' bo = kesM/resM· The above model is written as the form of the state space equation X,(t) = A,x,(t) +B,UI(t)
Wherein
B,
=
[~J
(13. 11) (13. 15)
Control Techniques for Ultrasonic Motors
Chapter 13
409
According to Fig. 13. 21, the controlled variable of the output in the MRAC is (13. 46) (2) The design of the reference model In the design of the reference model, the speed characteristic of USM should be considered sufficiently, but the capacity of USM cannot be exceeded. The reference model should satisfy the requirements of rapid response, a little overshoot, and no steady state error. According to the model reference adaptive control theory, the order of the reference model should not exceed the order of the system itself. After these factors arc considered, using the standard second order link as the reference model, whose state equation can be described as
{ xm : Am Xm Ym - CmX
+ Bm r
(13. 17)
where
A
= m
[ -
0 ktnl
Cm
=
[l,OJ
According to the requirement of the optimal response characteristic of the second order system, the damping ratio of the reference model is S"= o. 707, and the step response is designed based on the rising time t, = o. 14s, then
Am
=
[_
~61
_216. 8].
Bm
=
[3~lJ
(13. 48)
(3) The self-adapting law of the MRAC controller In the model reference self-adapting control system, the position error, which indicates the difference between the actual output of controlled object and the model output, called the generalized state error, that is (13. 19) From Eq. (13.19) ,we can obtain
x,(t)
=
[A, -B,F(t)Jx,(t) +B,K(t)Uf(t)
(13. 50)
From Eqs. (13. 17) and (13. 50), the equation of the generalized state error vector is (13.51) When
Ku
and F(t) are adjusted to
Ku =
K: and F= F* , respectively, the ad-
justablesystemmatehesthemodel. Here, Am=A,+B,F* and Bm=B,K: ,soEq. (13. 51) can be written as (13. 52) whereF=F* -Fm;K=K: -K u • The kernel in the design of the MRAC control system is to determine the selfadapting principle of Ku and F (t). Thus, a Lyapunov function within a space composed by general state error is defined
410
Ultrasonic Motors Technologies and Ap plicalions
(13. 53) where P, r-;1 and r;1 are the symmetric positive definite matrix. Obviously,
>
when em # O,L(t)
0
Differentiating two sides of the above equation to time, then Let)
=
1
~T
T··T
ZemPem+emPem+tr(F
r-;
1
-
-T
F+F
r-;
1 ~
~ e~ (PAm + A: P) em + tr(F T r-;1 F + x,e~
1 ~
---"
~
-1---"
F+Kr, K+Kr 2 K)
PBm K :
-1
F) (13. 54)
Because Am is set as a stable matrix during the design of the reference model, a symmetrical positive definite matrix Q can be chosen to establish PAm
+ AmP=
-Q. For arbitrary em #0, the first item of the above equation is negative definite. If choose
{F-
(B
---" = -
r
K
r, (BmK:
=-
Therefore, considering BmK: -1
=
1
B"
{F=
it
m
K'-l)Tp u
-1)T
emx,T
(13.55)
Pemur
then self-adapting law is
r
r
=
B: Pemx:
1
2
(13.56)
B: PemUr
Adjusting the parameters of the controller according to the above equation, the second and third items in Eq. (13.51) are both zero, and L(t) is negative definite. Obviously, the adapting theory designed by this way can guarantee the global asymptotic stability of the model reference adapting control system using USM. Thus, if t - CXJ, em (t) (1) The results of the control
90
r
"""
M
,...
....,
,...,
1""\
o.
,..,
- • - - OUlput of Ihe model OUlpul of Ihe molar
80
~60 ;;
g .'il 40 ~ 20
o o
10
15
20
25
lis (a) Response for asquare wave comm and
Fig. 13. 22
25 lis (b) Parameler F in square wave track ing "-
Results of MRAC using USM
Chapter 13
Control Techniques for Ultrasonic Motors
411
Figure 13.22 shows the results of MRAC control using USM. Where. Fig. 13. 22(a) is the result of the tracking control to a square wave when the reference input is 45 -90 It can be seen that the rotary speed of the motor tracks quickly the output of the reference model. and the tracking precision is relatively high. Fig. 13. 22 (b) gives the variation of the parameters of the self-adapting control. 0
13.4.6
0
•
Fuzzy Controller Using USM L2629 -
A fuzzy control is based on human's experience and knowledge. and employs the fuzzy reasoning as means. and makes decision through imitating the human's thinking manner to implement the technique of the computer intelligent control. The fuzzy control describes a system through some language variables. Its implementation does not need the precision mathematical model of a controlled obj ect. Therefore. control methods based on the fuzzy logic reasoning have become important control techniques using USM.
1. Fuzzy control law oj USM Figure 13. 23 shows the block diagram of a fuzzy position control system using USM. Its kernel is the fuzzy logic controller (FLC). which uses the two-dimension input structure. Generally. the two-dimension FLC takes the error of the controlled subject and the variance ratio of the error as the inputs. However. according to the operating feature of USM. in order to guarantee the high robustness of the control system when the parameters and the operating condition of the motor vary. the position error e (e = Yd - y) and the angular speed w, of the motor are taken as the inputs of FLC. The frequency adjusting method is used to regulate the speed of the motor. Therefore. a frequency base point f' is needed. which corresponds to the rated speed of the motor. In Fig. 13. 23. we use the increment of the operating frequency (frequency variable) 6.u as the output of FLC. and u as the control variable of the frequency converter. FLC
r-----------------------~
y.
+
S
e
AU
.~
-=
~
u(k-I)
I1r(k)
r f(k)
+
------USM Fig. 13. 23
Block diagram of fuzzy control system using USM
2. FuzziJication oj input/output variables Af ter the input/ output variables of FLC are determined. based on the requirement of the fuzzification. first of all. they need to be quantized. i. e.
412
Ultrasonic Motors Technologies and Ap plicalions
transforming them from a basic domain into the fuzzy domain. The grade of the quantization influences dramatically the control quality of the system. Generally. the smaller the quantization grades of the error e. the larger the system's overshoot. and the longer the transient process. Too much quantization grades will increase the calculation amount and the memory occupation. The main objective that this book uses the rotary speedw, of the motor as the input variable of FLC is to reduce overshoot. Theoretically. the thinner the partition of the quantization grade for w,. the smaller the overshoot. However. too slim partition will influence the response speed of the system. As the output of the FLC. too little quantization grades of 6.u will make the system oscillate. and too much quantization grades will elongate the transient process of the system's dynamic response. After considering these factors synthetically. we take the basic domain [ - 1 ( . 1( ] of the error e as fifteen grades. Introduce a sign X into the fuzzy domain of the position error
X= {-7.-6 ... ·.0.· ... +6.+7} We take the basic domain [ -10.10 J rad/ s of the rotary speed w, and the basic domain [ - O. 5. O. SJ kHz of 6.u as eleven grades; Y and Z indicate the signs of the fuzzy domain.
Y= {-S.-4.···.0.···.+4.+5}. Z= {-5.-4.···.0.···.+4.+S} During operating. the rotary speed w, of the motor may fall outside of [ -10. 10Jrad/s. then the quantization grades are set as 5 (when positive error) or -S (when negative error). The quantization factors of the position error e and the rotary speed w, arc G, = 9. 92(l/rad) and Gw=O. S5[ l/(rad/s)J. respectively. The proportional factor of the frequency variable is G/',u = O. l( kHz/l). During the controlling process. in order to increase the response sensitivity of FLC to little error. while quantizing the minor-error input signal, we properly increase the quantization factor. The language variables corresponding to e. w, • and 6.u are J:;. ~. and 6. l.l • respectively. The control strategy of the fuzzy control system using USM embodies in the fuzzy control rules. According to the operating characteristics of USM. the following language values for J:;. ~. and ~ l.l are introduced. {)JB (negative big). )JM (negative middle). )JS (negative small). ZO (zero). PS (positive small). PM (positive middle). PB (positive big)} Defining the fuzzy of variables is an important content of fuzzification. and its essence is to determine the shape of the membership function curve in the fuzzy domain. The membership grade functions defined by variables J:;. ~. and 6.1.1 arc shown in Fig. 13. 24. where J:; and ~ usc the triangle distribution function. and 6.U uses the single-point fuzzy set. Once the membership function curves are defined and discreted in the fuzzy domain. the membership grade of points in the fuzzy domain will be obtained. and the fuzzy subset of fuzzy variables arc constructed.
Chapter 13
:c!
Control Techniques for Ultrasonic Motors
413
0.5
"-
- 2 -I (a) Displacemcnls
NB
4 5
2
0
(b) Angular speeds
NS
NM
PM
PS
ZO
PB
~I
"-
I
o -5
-4
I
-3
I
-2
-I
I
0
I
I
I
2
3
I
5
4
(c) Frequency variable
Fig. 13.24
Membership degree functions of the input/ output variables in the FLC
3. Establishment of the fuzzy control rule The rule of the fuzzy control is based on the experience from manual control, and the rationality of its constitution directly influences the control quality of FLC. The experiments of the manual control using USM show that if the current position of the rotor is far away from the objective position, the current rotary speed of the motor is relatively slow, and then in order to implcment a rapid positioning, the motor needs to speed up. Therefore, in order to decrease the driving frequency of the motor, FLC needs to export a negative frequency-vary value. On the contrary, if the current position of the motor is elose to the expected position, and the speed of the motor is relatively fast, the motor needs to decelerate. Thus, need to increase the driving frequency of the motor, FLC needs to export a positive frequency-vary value. According to these experiences obtained from the operating, the fuzzy control rules for USM are established as shown in Table 13. 2. Table 13. 2
Fuzzy control rule for USM E ~
fill
W
ZO
PS
PM
PB
NB
NM
)IS
NB
ZO
PS
PM
PB
PM
PS
ZO
NM
NS
ZO
PS
PM
PS
ZO
)IS
NS
NM
NS
ZO
PS
ZO
NS
)1M
ZO
NB
NM
)IS
PB
NS
NM
)lB
PS
NM
NS
ZO
PS
ZO
NS
)1M
PM
NS
ZO
PS
PM
PS
ZO
)IS
PB
ZO
PS
PM
PB
PM
PS
ZO
I
414
Ultrasonic Motors Technologies and Ap plicalions
In order to carry out fuzzy reasoning, the fuzzy control rules from above table should be written as the fuzzy condition proposition, that is Rule 11 if I.:; =:'\JB and ~ = NB then L l.l = ZO; Rule kj Rule 77 where,1k
W=B
and if I.:;
,/.J) ,
=
then
~)
L l.l
=
!;;k) ;
then L l.l = ZOo PB represent the language values of I.:;, ~, and Ll.l in their do-
PB
and C)
~
and
~=
main, respeeti vcly. According to the fuzzy mathematical theory, every fuzzy condition sentence corresponds to a fuzzy relationship, that is
ISk) = ,1k X 12) X !;;k)'
k = 1, 2 , ... , 7; j = 1, 2 , ... , 7
(13. 57)
All of these control rules correspond to a general fuzzy relationship IS, which can be expressed as
IS
k=7,j=7
=
U ,1k
k=l,j=l
X
12j
X
!;;kj
(13. 58)
The membership function of IS can be calculated according to the following equation /1R (.:r:,y,z)
i=7,j=7
=.
V
I-l,)-l
/1A, (.:r:)
1\
/1B, (y)
1\
/1c, (z) ,
.:r:E X,yE Y,zE Z
(13.59)
where /1::t, (.:r:) is the membership grade of the fuzzy subset ~k at the position error x;/1':c (y) is the membership grade of the fuzzy subset ~j at the angular speed y; /1c.. (.:r:) is the membership grade of the fuzzy subset
fk) at the frequency
variable z. when the fuzzy values of the position error and the angular speed inputted to FLC are ~' and ~' respectively, according to the fuzzy composition rules, the fuzzy value of the frequency variable exported by FLC is (13. 60) where" 0" is the fuzzy synthesis operator. The above equation can be written as the form of the membership function: (13.61)
4. DeJuzziJication The frequency-variation value obtained through the fuzzy reasoning equation is a fuzzy vector, and cannot be directly used for the control using USM. So, we need to decide an explicit value from this vector, and carry out the dcfuzzifieation. In all kinds of dcfuzzifieation methods, a center of gravity (COA) method is simple, and easy to be implemented, and its compositive performance is good. Therefore, the explicit value of the frequency-variation variable using COA method is
Control Tcchniqucs for Ultrasonic Motors
Chaptcr 13
415
II
~ fl.;:,u (Zi) Zi LU
=
Zi
...:i----;I7-\- - - -
E Z
(13. 62)
~fl.;:,u (Zi) i-"j
where fl.;:,\[ (zJ is the membership grade of the frequency variable L l.l in its fuzzy domain. Theoretically. according to the value of e and w, gathered. using the Eq. (13.61) and Eq. (13. 62). the corresponding grade of frequency variable LU can be calculated online. However. in order to improve the real-time behavior of the control, we can calculate off-line the frequency variable LU according to all possible combination of the elements in X and Y. and then make a look-up table of the fuzzy control using USM. as shown in the Table 13. 3. and store this table in the system. When the fuzzy control using USM is needed. the computer looks up this table according to the position error and the angular speed of the motor. therefore. we can obtain the frequency-vary value needed within the current sampling period. Table 13. 3 t,U
w -4
-5 1
E
Fuzzy inquiry control table of position control using USM
-2
-3 1
1
-1 1
0 1
+1 1
+2 1
+3 1
+4 1
IH
-7
0
-1
-1
-2
-3
-5
-3
-2
-1
-1
0
-6
0
0
-1
-2
-3
-5
-2
-2
-1
0
0
-5
1
0
-1
-1
-2
-1
-2
-1
-1
0
1
-4
1
0
0
-1
-2
-4
-1
-1
0
0
1
-3
1
1
0
-1
-1
-3
-1
0
0
1
1
-2
2
1
1
0
0
-2
0
0
1
1
2
-1
3
2
1
1
0
-1
0
1
1
2
3
+0
5
1
3
2
1
5
1
2
3
1
5
+1
3
2
1
0
0
-1
0
0
1
2
3
+2
2
1
1
0
0
-2
0
0
1
1
2
+3
1
1
0
-1
-1
-3
-1
0
0
1
1
+1
1
0
0
-1
-2
-1
-2
-1
0
0
1
+5
1
0
-1
-1
-2
-4
-2
-1
-1
0
1
+6
0
0
-1
-2
-3
-5
-3
-2
-1
0
0
+7
0
-1
-1
-2
-3
-5
-3
-2
-1
-1
0
Actually. the value obtained from the fuzzy decision is only a grade in the output fuzzy domain. and is not the controlled value requested by USM. Therefore. it needs to be multiplied to a proportion factor G;:,u to become an accurate value. In addition. FLC output needs to add to the control value u(k - 1) obtained from
416
Ultrasonic Motors Technologies and Ap plicalions
the last sampling period to become the control value of the current sampling period, because Fig. 13. 23 uses the increment structure for the controller. That is (13. 63)
5. Control results In order to inspect the control effect of FLC, we made the fuzzy control experiments of TRUM-60 on the control system established above. Set the reference frequency as 40kHz. Considering that the fuzzy control determines the control value by inquiring the table, the sample period is set to 4ms. Fig. 13. 25(a) shows the response curve of the system in 90° step input, its stable error is within ±O. 28°, the response time is around o. 18s, and the control precision is low. Fig. 13. 25 (b) shows the step response curve under load. It can be seen that the response time increases a little (about o. 2s) after load addcd, but the control error docs not change much. This indicates that the fuzzy control has relatively high robustness to load changes of the motor. 90 80
""
" 00
60
" .~
40
'"
20
:a 0
100
~
""
" 00 :a
50
" .~ 0
'"
0 0
0.5
1.5
0
2
/ 0 0.2
0.5
1.5
2
tis
tis
Ca) Step response without load
Cb) Step response with O.2N·m load
Fig. 13. 25
Step response of fuzzy position control
Figure 13. 26 shows the tracking results of the USM with 90° square wave input. It can be seen that USM oscillates within 0-90° repeatedly and carries out the position tracking rapidly when the target position varies within 0-90° according to the pattern of the square wave. This indicates that the fuzzy control
,
""a'"' 00
90 80
I
60
rr II
I
I
"
(5
'"
20 0
I
, 2
4
I
I
6
8
Actual value
_ . Ex pected
, I ,
I
o
-
,I
C 40
f:a
.n
I
I
I 10 12
value
~~ I
I
I
I 14
"
0"
12
'ED
.
~
,." ::
I
I
16 18
20
"2 "§
U
1 000
500
0 - 500 -1 000 ~-L~-L~~-L~LL-L~-L~LL~
o
2
4
6
8
~
(a) Square wave tracking
10
12
14
16
~
(b) Varial ion of cOl1lrol qualll ily
Fig. 13. 26 Square wave tracking results of fuzzy position control using USM
18 20
Chapter 13
Control Techniques 10r Ultrasonic Motors
417
can implement a rapid position tracking control using USM. It is worth to be noticed that the control value in Fig. 13.26 (b) is the digital format of a frequency control variable, and when its value is negative, "negative sign" only indicates the control effect which makes the motor to rotate reversely.
References [ 1
J
Chunsheng Zhao. Recent progress in ultrasonic motor techniques. ] oumal of Vibration, Measure&. Diagnosis, 2004, 24(1): 1-5. (in Chinese)
ment
[ 2
J
Jiakui Zu. Research on Driving and Control Techniques for Traveling- Wave Ultrasonic Motor Based on Its Electric Characteristics. Post-doctoral Report. 'lanjing: 'lanjing University
o[ Aeronautics and Astronautics, 2004. (in Chinese) [ 3
J
Huafcng Li. Study on Ultrasonic Motor and Its Precise Servo-control System. Dissertation for the Degree of Doctor of Philosophy. Wuhan: Huazhong University of Science and Technology, 2002. (in Chinese)
[ 1
J
[ 5
J
Honglin He. Research on the Ultrasonic Motor and Its Application in the Robot. Dissertation for the Degree of Doctor of Philosophy. 'lanjing: 'lanjing University of Aeronautics and Astronautics, 2007. (in Chinese) M W Spong. Robust and adaptive control o[ manipulators. IEEE Transactions on Automatic Control, 2001(3): 186-210.
[ 6
J
[ 7
J
S Furuya. Load-adaptive frequency tracking control implementation of two-phase resonant invert [Dr ultrasonic motor. IEEE Transactions on Power Electronics, 1992, 7(3): 542-550. T Seniyu. Adjustable speed control o[ ultrasonic motor by adaptive control. IEEE Transactions on Power Electronics, 1995, 10(5): 532-538.
[ 8
J
T Senjyu, T Kashiwagi, K Uezato. Position control of ultrasonic motors with adaptive deadzone compensation with fuzzy inference.
IEEE Transactiuns un Power Electrunics, 2002,
17(2): 265-272. [ 9
J
J Mass, T Schulte, "I Frohlcke. Model-based control for ultrasonic motors. IEEE/ ASME Transactions on Mechatronics, 2000,5(2): 165-180.
[10J
Sahin Yildirim. Design o[ adaptive robot control system using recurrent neural network. International ] oumal of Intelligent and Robotics System, 2005, 11(3): 217-26l.
[llJ
T Senjyu, H Miyazato, S Yokoda, et al. Position control of ultrasonic motors using neural
[12J
network. IEEE Transactions on Power Electronics, 1998, 13(3): 381-387. Faa-jeng Lin, Rong-J ong Wai, Rou- Yongi Duan. Neural-network controller [or parallel resonant ultrasonic motor drive. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 1999(4): 494-501.
[13J
Faa-jeng Lin, L C Kuo. Identification and control o[ rotary traveling-wave type ultrasonic motor using neural-networks. IEEE Transactions on Control Systems Technology, 2001(4): 672-680.
[I1J
Y Izuno, R Takeda, M Nakaoka. 'lew fuzzy reasoning-based high-performance speed/position servo control schemes incorporating ultrasonic motor.
IEEE Transactiuns un Industry
Applications, 1992, 28(3): 613-618.
[I5J
G Bal. A digitally control drive system for traveling-wave ultrasonic motor. Turkey] ournal Electrical Engineering and Computer Sciences, 2003,11(3): 155-167.
[16J
S W Chung. A thesis submitted in partial [ulfillment o[ the requirements [Dr the degree o[ master of philosophy. Motion Control of a Traveling-wave Ultrasonic Motor. Hong Kong: University of Hong Kong, 200l.
[17J
S W Chung, K T Chau. Servo speed control o[ traveling wave ultrasonic motors using pulse width modulation. Electric Power Components and Systems, 2001, 29(8): 31-37.
[I8J
Faa-jeng Lin, Rong-Jong Wai, Hsin-Hai Yu. Adaptive fuzzy-neural-network controller for
418
Ultrasonic Motors Technologies and Ap plicalions ultrasonic motor drive using LLCC resonant technique. IEEE Transactiuns un Ultrasunics,
Ferroelectrics, and Frequency Control, 1999(3): 715-727.
[l9J
Faa-jeng Lin, Rong-jong Wai, C C Lee. Fuzzy neural-network position controller for ultrasonic motor drive using push-pull DC-DC converter. IEEE Proceedings on Control Theory Application, 1998, 34(1): 363-368.
[20J
Zhihua Chen. Research on the Control of Ultrasonic Motors. Post-doctoral Work Report. Nanjing: Nanjing University of Aeronautics and Astronautics, 2003. (in Chinese)
[21J
Zhihua Chen, Chunsheng Zhao. The control of resonant tracking of ultrasonic motor. & Acoustooptics, 2003, 25 (2): 119-151. (in Chinese) Honglin He, Hua Zhu, Chunsheng Zhao. Position control of an ultrasonic motor using fuzzyPI technique. Mechanical Science and Technology, 2006, 25 (5): 603-607. (in Chinese) Piezoelectrics
[22J [23J
Shouren Hu, Dewen Hu, Chun Shi. Application of Neural Networks. Changsha: National University of Defense Technology Press, 1993. (in Chinese)
[24J
Shousong Hu. Theory of Automatic Control (4th Edition). Beijing: Science Press, 2001. (in Chinese) Huafcng Li, Chenglin Gu. Precise position control of ultrasonic motors using adaptive control. Piezoelectrics & Acoustooptics, 2003, 25(2): 155-158. (in Chinese)
[25J [26J
Huafeng Li, Chunsheng Zhao, Chenlin Gu. Precise position control of ultrasonic motor by fuzzy control. Journal of Huazhong University of Science & Technology, 2005, 32 (5): 22-21. (in Chinese)
[27J
Honglin He, Chunsheng Zhao. Fuzzy-neuron network control techniques applied to a robot driven directly by ultrasonic motors. Piezoelectrics &. Acoustooptics, 2006, 28 (2) : 143-146. (in Chinese)
[28J
Shiyong Li. Theory of Fuzzy Control, Neural Networks and Intelligent Control. Harbin: Harbin Institute of Technology Press, 1996. (in Chinese)
[29J
Zhenzhong Dou. Theory and Applications of Fuzzy Logic Control. Beijing: Beijing University of Aeronautics and Astronautics Press, 1995. (in Chinese)
Chapter 14
Testing Techniques for Ultrasonic Motors In addition to theoretical research on ultrasonic motors, we also investigate their tcsting techniqucs. Testing can not only verify the theoretical method, but also propose solutions for some problems that can not be resolved by theoretical analysis. Furthermore, the performance of the products also requires e.Lperiments according to certain standard measurement methods. Therefore, much attention has been paid to the testing techniques of USMs around the world .1 " . With the broad applications of USMs in aerospace field, semiconductor manufacture field, etc., the USA and Japan conducted certain basic research on USMs under extreme environmental conditions in 1998. Russia and Germany conducted some explorations in this aspect as well. Unfortunately, these techniques and data have not been reported C' 7:. In 2004, the European Space Agency successfully developed USMs suitable for vacuum environments and conducted many experimental studies, ineluding mechanical characteristics, life testing, and vacuum thermal cycling experiments. At present, research institutes in China have also been carrying out some preliminary research on testing techniques for USMS·8-11 .. USM testing techniques mainly include modal testing of parts and assemblies, measurement of pre-pressure, transient characteristics, mechanical performance, adaptivity to extreme environment conditions·]5J, their lives, etc. USMs run at a high frequency above 20kHz but low speed, contrastively, traditional electromagnetic motors(EMs) usually run at a low frequency of 50Hz/60Hz but high speed. Thus, many measuring devices and methods for traditional electromagnetic motors cannot be directly used for USMs. The mechanical characterization measurement system for EMs, including the hardware and software, must be updated. This chapter mainly describes the purposes, requirements, methods, equipment and results of the testing for USMs.
14. 1
Modal Testing for Parts and Assemblies
An ultrasonic motor USM relies on the converse piezoelectric effect of piezoelectric ceramics to excite the resonance of stator in ultrasonic frequency range and realize its rotation. Therefore, the natural modes of the stator are critical to the performance of the USM. It is always e.Lpected that one of the modes of the manufactured stator can be consistent with pre-calculated design, and greater response can be obtained under a certain e.Lcitation voltage. Furthermore, in order to confirm the design of the stator and check the manufacturing precision,
C. Zhao, Ultrasonic Motors © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011
Ultrasonic Motors Technologies and Ap plicalions
420
the correctness of the piezoelectric ceramic bonding process and to guarantee the assembly quality, the modal test for USM is necessary. Traditional modal tests employ some special excitations (such as hammering) to excite a structure and get its modal response. The response is then converted into electrical signals by using contact-type vibration sensors. After the signal amplification and data processing. the modal parameters of the structure are acquired 116_. Considering that the USM's stator is a structure with small size and light weight, the use of the general contact-type sensors will have non-negligible effect on the modal characteristics of the stator. Therefore, in the initial stage of the author's study, due to the test equipment restriction, we used dynamic signal analyzer HP3562 to measure admittance curve of the stator and obtain its modal frequencies. With the aid of laser holography, we can measure the nodal pattern of the stator. Then, SS330 type electronic speckle laser vibrometer is used for measuring modal frequencies, and node patterns can be measured at the same time 1l7 -. The above apparatus and measurement methods have played an important role and helped us to complete a series of experimental studies 1l8- 19 -. However, electronic speckle laser vibrometer is only suitable for measuring the node pattern of out-plane vibration, and is not capable of measuring longitudinal vibrations and in-plane vibrations of bar-type stator as well as the quantitatively measuring amplitudes. Meanwhile, its adjustment and operating are more troublesome, and it is susceptible to the external environment interference. These disadvantages confine the application to electronic speckle laser vibrometer in the USM studies. In 2002, we adopted a more advanced device PSV-300F-B type Doppler laser vibration measurement system (PSV-300F-B) :20 J , as shown in Fig. 14. 1. The system has some advantages, such as wide test band (0-100kHz), a high accuracy (displacement can be measured to nm level) and non-contact measurement. etc. It can achieve a rapid multi-point measurement through the definition of scanning grid on the measured stator, and then obtain quantitatively amplitude vs. frequency curve, phase vs. frequency curve. and the mode shape of the stator.
Fig. 14. 1
PSV-300F-B type Doppler laser vibration measurement system
Chapter 11
Testing Techniques for Ultrasonic Motors
421
The block diagram of PSV-300F-B is shown in Fig. 11. 2, and the operating principle of the system is shown in Fig. 14. 3. It can be seen that the system consists of two major components hardware and data-processing software. The high-precision laser interferometer is the core of the hardware. In the system, the computer scanning module produces a digital signal, which is converted into an analog signal by D/ A converter in connecting box. The analog signal is amplified into an excitation voltage by a power amplifier. The voltage is applied to a target tested, such as a stator, finally then induces the target to vibrate. At the same time, a laser beam. which comes from the laser interferometer in the scanning laser head and has a certain frequency. irradiates the surface of the measured vibrating target, and the reflected scattering laser from the measured target is collected. The reflecting light beam induces the certain frequency to change due to the Doppler effect and interferes with the laser beam (as reference) in scanning laser head, then generates the Doppler frequency shift which is proportional to the vibration velocity of the measured target. Photoelectric detector records the interference signals and outputs an analog voltage signal through the decoder processing. This signal is processed by the computer via a high-speed A/ D conversion in the connecting box, and finally mode shapes and amplitude fre-
Output module Animation module Driver module
Scanning module Analysis module -----------------------------------------~
Computer control and date processing system
Fig. 14.2
System hardware
Components of PSV-300F-B type Doppler laser vibration measurement system
Controller
-
..... - -----
synchronillion conncclor -
~.~ Vu ,•. ~~I---"' U M slator
Fig. 14.3
-- --
-
1'VV
. -~
1
Power amplifi_
................ -
Schematics of PSV-300F-B type Doppler laser vibration measurement system
Ultrasonic Motors Technologies and Ap plicalions
422
quency response curves can be recorded and displayed on the computer monitor. In this system, the scanning and measurement of the stator surface are completed through the couple of high-speed swing lens in the front of the laser interferometer. Software and computers arc used for the automatic control, data quality assessment, modal analysis, and display of the full system. The modal testing of TRUM-60 stator shown in Fig. 11.1 is carried out by PSV-300F-B and its measurement results are shown in Figs. 11. 5-11. 7. The stator frequency response characteristic curve is shown in Fig. 14. 5. While using PSV-300F-B the measuring beam must be directed to the target surface (such as the upper surface of the stator), generally it is used for the measuring of frequency response of the stator without the rotor. Figs. 14. 6 and 14. 7 show B09 mode shape and nephogram of the stator, respectively. From the two figures we can get the relative and absolute velocities, amplitude and node line position. It can be seen from Figs. 14. 6 and 14. 7 that the stator's mode B09 possesses 9 nodal diameters and its velocity amplitude reaches O. 8m/ sunder 38. 2kHz. The test results arc in good agreement with the calculated results through the theoretical model in Chap. 5.
~
..,~"
15
.~ 10
Q.
"OJ
.g
5
~
0
l"'-_. . . ~~_. . . .;: : .J:. . ,.: '"-~
+--__ W
~
~\..~
__
50
~
J IkHz
Fig. 14.4
Fig. 14. 5 Frequency response of stator of TRUM-60
Stator of TRUM-60
Fig. 14. 6 Modc shapc stator of TRUM-60
B09
of
Fig. 14.7 Mode nephogram of TRUM-60
B09
of stator
Some types of USMs operate based on the in-plane mode of a stator. For ex-
Chapter 11
Testing Techniques for Ultrasonic Motors
423
ample. a longitudinal-torsional type USM is designed by utilizing the longitudinal and torsional modes of the stator. and the torsional mode is exactly an in-plane mode. The measurement principles of the in-plane and out-plane modes are similar, although the measuring devices are different. The more details can be obtained in Refs. [21-26].
14.2
Measurement of Pre-pressure
Pre-pressure refers to the pressure between the stator and rotor in assembled products. It has an important influence on the dynamic characteristics of the stator, the contact characteristics between the stator and rotor, and whole USM mechanical performance. The appropriate pre-pressure can effectively reduce the abrasion and noise. and ensure that the whole system has good output performance. The relationship between USM performance and pre-pressure can be obtained through pre-pressure testing. so we need to design the pre-pressure range of USM and ensure that USM has excellent performance and a longer service life. The pre-pressure testing mainly includes measurement of pre-pressure, the relation between USM's performance and the pre-pressure, etc. Since there are no commercially available test devices, PDLab designed and manufactured several sets of the pre-pressure measurement devices for TRUM series products. A new pre-pressure test device was developed in order to investigate the influence of the pre-pressure on the USM's performance[28:. The schematic picture and measurement system of the pre-pressure are shown in Figs. 11. 8 and 11. 9, respectively. In Fig. 14. 8. the pre-pressure between the stator and rator of USM is applied with the load rod of pressure sensor. USM
Fig. 14. 8
Pressure sensor
Pre-pressure regulator
Pre-pressure adjusting device
Fig. 14.9
Pre-pressure measurement system
The pre-pressure regulator supporting plate is designed to "II" shape, which avoids the radial slide of the sensor and ensures that the sensor has only one axial degree of freedom. This can also guarantee the pre-pressure measurement accuracy. Moreover, clearance fit is adopted between the supporting plate and the inner hole of the shell, and their contact areas and sliding friction are reduced because of the "II" shape design. The regulator with fine-pitch threads can make pre-pressure to adjust precisely. The motor and the sensor are assembled together, and the latter is used for picking up the pre-pressure, which is sent to the measurement system. The pre-pressure value can be automatically adjusted and
424
Ultrasonic Motors Technologies and Ap plicalions
displayed by the system. Figure 14. 10 is the measurement results of USM using the new pre-pressure regulator. It proves that pre-pressure has great influenee on the mechanical characteristics of the USM. When the input voltage and excitation frequency remain the same, different no-load speed and stall torque are obtained under different pre-pressure respectively. From Fig. 14. 10, there is an optimized range of the pre-pressure, which can make no-load speed and stall torque to elose simultaneously to the maximum value. 1.4 r - - - - - - - - - - - - - - - - - - --,
250r-----~----~----~----~----__,
L2
200
EI
c:
~
150
""!:l
100
>::' Q.
~0.8
g
0.6
f-
50
0.4 0.2
100 Pre-pressure forceIN
Fig. 14. 10
14.3
200
300
400
500
Pre-pressure forceIN
USM's pre-pressure vs. no-load speed and stall torque
Measurement of Transient Characteristics
A USM has very important characteristics, which are the transient characteristics in startup and shutdown processes. Two main parameters can be used to describe the transient characteristics-startup and shutdown response time. Compared with traditional electromagnetic motors, the response time of the USM is very short (m:,~level). It means that the startup response time from zero speed to the steady speed or the shutdown response time from the steady speed to zero speed is shorter. The response time is very important indicators for an USM being used. Therefore, it is very necessary to conduct transient characteristics testing for USM and to measure accurately its transient response time. The author has conducted theoretical and experimental research on the testing methods of the transient characteristics: 29 3OJ. Since USM is a new product, there is no commercialized system to meet the measurement requirements for transient experiments. Thus, the author designed the system, which can basically meet those of the transient experiments·"·"J.
14. 3. 1
Testing Principle
1. Hardware implementation The hardware of a typical transient characteristics testing system consists of an optical encoder, timer/counter, controller and computer. The schema of the system is shown in Fig. 14. 11. The optical encoder converts the angular position of USM into a standard electric pulse signal. A data acquisition board uses its own
Chaptcr 11
Tcsting Tcchniqucs for Ultrasonic Motors
425
crystal oscillator to measure the output pulse width of the optical encoder. One pulse width corresponds to the time spent by the optical encoder when it rotates through a certain angle, so the average speed of the motor at this time can be obtained. The controller controlled by the computer can start and stop the timer/ counter according to instructions.
Optical encoder
Fig. 14. 11
Transicnt charactcristics tcsting systcm for USM
The following is the velocity measurement principle of a SZGM-Ol incremental type (photovoltaie) encoder. The output signal indicator of the SZGM-Ol encoder is 5 000 p/ r. that is, if a motor drives the shaft of optical encoder through 360 the optical encoder will output 5 000 square wave pulses and each pulse corresponds to o. 072 0 angular displacement. Let n be the motor speed (r/s), and then the time corresponding to each pulse width is 0
,
t =
o. 0002/n
(14. 1)
The 8 253 counter uses a counter clock input from a board clock (1MHz) and a gated trigger signal, and its operating manner is o. The time corresponding to each pulse of the optical encode can be calculated as below: _ 2 65536(k t-
+ 1) -
(HB X 256 + LB)
f
(14. 2)
where k is the number to reach the full initial value of the counter, HE is the high byte of the initial value of the timer/ counter 0 with a range of 0 to 255, LE is the low byte of the initial value of the timer / eoun ter 0 with a range of 0 to 255. and f is the input clock frequency of the timer/ counter o. The average speed within each pulse width time can be obtained from the previous two equations:
n=
10000[65536(k
f
+ 1) -
(HE X 256
+ LE)]
(11.3)
2. Software implementation The software of the transient characteristics testing system has be developed based on Visual C++6. 0 platform. The stop point of the counter in startup response testing can be determined according to the given data. but the zero point of the speed in shutdown response testing can not be determined in advance in the testing process. Therefore. we adopt the following approach: the counter keeps on cycling until the motor stops and the data is reorganized by using memory copy technique.
426
Ultrasonic Motors Technologies and Ap plicalions
14. 3. 2
Transient Characteristics of USMs
In order to analyze and quantitatively assess transient speed characteristics, at first, we define an average speed, and on this basis, we regard the startup and shutdown responsc timcs as two main transient charactcristic indicators. no describcs the absolutc avcrage spccd of motors under a ccrtain condition and within a certain time interval. The average speed of a motor is defined as below:
1
no =
N
N~ni
.1
(11. 1)
i= l
wherc N is the number of sampling points in a wholc cyele (except the startup and shutdown process); n i is an instantaneous specd corresponding to every sampling point. Averagc spced rcflects a stable opcrating spced of thc motor under a certain operating frequency. voltage and current.
1. Startup response time The startup rcsponsc time is defined as below: (14. 5)
In this cquation,
tql
is thc momcnt at which 95 % of the averagc specd is achicvcd
after thc motor is started; and tq2 is the moment at which 5 % of the averagc spced is rcachcd after the same process. The startup response timc is not only relatcd with motor types and their structural forms but also their driving voltages and frequencies.
2. Shutdown response time The shutdown response time is defined as below: (11.6)
wherc
tgl
is thc moment at which 5 % of thc avcrage spccd is achicvcd after thc
motor shuts down; and t g , is thc momcnt at which 95 % of thc averagc spccd is reached after the same proccss. Shutdown response timc is not only relatcd with motor types and structural forms but also their driving voltages and frequencies. Thc startup and shutdown charactcristics for TRUM-30 measured in thc tcsting are shown in Figs. 11. 12 and 11. 13, respectively. 100
Swtllfl llnqllHl 29.01
il 52
'\~s~ll mJIII
~
Enter
~ 26 o
~"
75
"!l ~ c g
50
{?
HIl561
C"",,,I
'"
'"
120
240
360
400
600
Ilms
Fig. 14. 12 Measured startup characteristics for TRUM-30
720
CboorT.,q_·1 ,\\",. ,plXdll •• )
0'1-1)
9WO)
Enter
Cancel
25
0
0
12 1
242
363
464
605
Ilms
Fig. 14. 13 Measured shutdown characteristics for TRUM-30
726
Chaptcr 11
14.4
Tcsting Tcchniqucs for Ultrasonic Motors
427
Measurement of Load Characteristics
USM has the same load characteristics as traditional electromagnetic motors, mcluding speed vs. torque, efficiency vs. torque and output power vs. torque, and so on, which are the most important indicators of USM's performance. During the design of USM, although the load characteristics can be estimated according to its dynamic model, these characteristics still have to be experimentally verified ultimately. In order to obtain the load characteristics, PDLab bought a set of automatic testing system for load characteristics. Since this system is optimized for the general high-speed AC-DC motors, but it cannot be directly used for measuring the characteristics of the low-speed USM excited by AC voltage in ultrasonic frequency range. Therefore, both the hardware and software of the testing system have been updated to increase the measurement accuracy. The improved equipment has the following features: (1) It can automatically measure the USM's load characteristics within one minute, including the speed vs. torgue load, output power vs. torgue and efficiency vs. torque, etc.; (2) Torque measurement range is from O. 001:'\1'm to 5N'm; (3) The testing results can be displayed on computer screen or printed.
14. 4. 1
Measurement System for Load Characteristics
Figure 14. 14 shows load characteristics measurement system for USM. It mainly consists of a hysteretic dynamometer, DW-7 AC power instrument, DW-8 DC power instrument, KD-7 dynamometer display instrument, M2020 dynamometer controller and computer. For the hysteretic dynamometer, we have type CC500, CC5K and CC50K to be chosen according to the requirements of the measured torque and speed of the motor.
Fig. 14. 14
Measurement system for USM load characteristics
The data of the voltage, current and input power of the measured motor are sent to the computer via a RS-232C interface. The data of the torque, speed, output power are transmitted to the computer via the RS-232C in the dynamome-
428
Ultrasonic Motors Technologies and Ap plicalions
ter controller. The computer with human-machine operating interface acqUlres data and issues control signals through RS-232C.
In an automatic testing
process, the dynamometer controller sends control signals to the hysteretic dynamometer to complete an automatic loading process. The hardware connection relations within the testing system arc shown in Fig. 14. 15.
I=:::CO===1
Hysteretic dynamometer
Dynamometer controll er
Fig. 14. 15
Block diagram of measurement system for USM load characteristics
1. Hysteretic dynamometer The system with various types of the hysteretic dynamometers can meet different measurement requirements for different kinds of USMs. One hysteretic dynamometer consists of a hysteretic brake. torque sensor and speed sensor. The measured motor drives a ring-shaped rotor in the hysteretic dynamometer. After a magnetizing current is applied to the brake, the inner and outer stator of the hysteretic dynamometer form an air-gap magnetic field, which imposes a braking torque on the measured motor through the ring-shaped rotor. The torque is converted into a voltage signal by the torque sensor. Meanwhile, the motor speed is converted into electric pulse signal by the optical encoder. Then, we can obtain the operating parameters of the motor.
2. DW-7 digital power analyzer The DW-7 digital power analyzer continuously samples the voltage and current of the measured motor under the control of the mieroeontroller and calculates its input power. The real-time voltage, current and input power of the measured motor arc displayed on the screen. and these values can be locked or released by using a "HOLD" key. At the same time, the measured values can be sent in real time to the computer through the RS232 interface.
3. KD-7 dynamometer display The KD-7 dynamometer display continuously samples the electric signal provided by the torque sensor of the dynamometer and the electric pulse signal of the measured motor speed under the control of the micro controller , and the computer calculates the output power of the measured motor. The real-time torque, speed and output power values of the measured motor arc displayed on the screen, and these values shown can be locked or released by using a "HOLD"
Chaptcr 11
Tcsting Tcchniqucs for Ultrasonic Motors
429
key. At the same time. the measured values can be sent in real time to the computer through the RS232 interface.
4. M2020 Dynamometer Controller The M2020 dynamometer controller is the control interface of this system for automatically testing. When the system is in operating. it adjusts automatically brake current and maintains a stable motor speed under the control of the mierocontroller and phased-locked loop circuit. It establishes the communication between the RS232 and computer as the rotor speed is stable. The motor torque and speed values arc sent to the computer for processing through the RS232 mterfaee of the system.
14. 4. 2
Measured Results for TRUM-60 Load Characteristics
The software used for the load characteristics measurements consists of four modules: document management and print module. measuring module, testing module and display module. According to the motor type and the highest speed, the computer can automatieally load the torque, collect data, and display all load characteristics curves, as shown in Fig. 14. 16. 100
250
20.0
90
225
18.0
80
200
70 ~
}
ii
·u E W
16.0
Output power
175 .:;:
14.0
150
12.0
125
10.0
100
8,0
30
75
6,0
20
50
4.0
10
25
2,0
60 50 40
0
~
~<>
"a. '"
0
0
Fig. 14. 16
0.8
1.2 Output torq ucl(N· 111)
0,0 1.6
~ "
"'"'0
.§.
8
2.0
Load charactcristics for TRUM-60
In Fig. 11. 16, the load characteristics of the torque versus speed, output power and efficiency are shown. The motor speed decreases with the increase of the torque, and both the output power and efficiency are the quadratic functions of the torque, which is basically identical to the prediction results of the theoretical model (see Cha p. 5).
430
14.5
Ultrasonic Motors Technologies and Ap plicalions
Environmental Testing for Ultrasonic Motors
As a new type of driving devices, USM has many unique advantages and wide application prospects in aerospace and semiconductor manufacture fields, etc. In these applications USMs have to bear rigorous environmental conditions, such as violent vibration, strong impact, high/low temperature and vacuum, etc, which are quite different from normal environmental condition. Consequently, it is necessary to study the operating characteristics of USMs in above harsh conditions. At present, our research is on the excessive process from laboratory to commercialization. Research on the environmental testing of USMs is also one of the bottlenecks for its industrialization. We must establish a set of testing equipment and some methods for environmental experiment in order to obtain the relevant testing data and improve product reliability of the USMs in the extreme environment condition.
14. 5. 1
High/Low Temperature Environmental Testing
The temperature environment in the aerospace field is very harsh. For example, the environmental temperature range of JWST (James Webb Space Telescope) operating is from -243°C to +460°C, and that of ='JASA Mars probe operating can reach + 160°C. In addition, the temperature range of the Moon surface is from -130°C to 180°C. ]PL and MIT had carried out a series of environmental testing including high and low temperature environmental experiment in order to apply USMs to micro lander for Mars detection. In order to put USMs into actual applications especially into the aerospace field, it is necessary to explore methods for USM high/low temperature testing and to establish testing devices. According to the standard testing requirements of electronic products in the high/low temperature environment, the author proposes following conditions for the environmental testing of USMs in high/low temperature:
1. High temperature environment testing Before testing, the appearance of USM sample must be inspected, and electrical and load characteristics must be checked. An optional testing sample without packaging and electricity at room temperature is put into test chamber. The chamber's temperature can be controlled to the required temperature. The increasing rate of the temperature inside the chamber is no more than 1°C / min. The testing temperatures can be adjusted to 40°C, 55°C, 70°C, 85°C, 100°C, 125°C, and 155°C, respectively, and the each temperature is maintained for 30 minutes. At each testing temperature, the USM is applied by power and mechanical load to observe the load characteristics. After the high temperature testing is finished, the USM sample without load can be put into the standard atmosphere environment for 8 hours, and its load characteristics will be measured once agam.
Chapter 11
Testing Techniques for Ultrasonic Motors
431
2. Low temperature environment testing The testing requirements and proeess for the low temperature environment are similar to the high temperature testing except that the testing temperatures are S'C. -S'C, -10'C, -ZS'C. -10'C. -S5'C. and -6S'C. 3. Testing System A high/low temperature environmental testing system consists of a high/low tcmperature tcst chamber, tcstcd USM and specd/ torquc measuring instrumcnt. The USM is loadcd by the spccd/torquc measuring instrument, which measures its speed and torque under the high/low temperature environment at the same time. The testing device is shown in Fig. 11. 17.
Speed/torque measurement instruments
Speed/torque monitor
(a) Test system
Fig. 14. 17
(b) Photo of testing system
High/Low temperature testing system for USM
In Fig. 14. 17, a tcmperaturc scnsor used for monitoring thc test conditions is installed in the high/low temperature chamber. In order to maintain the uniformityof thc tcst conditions. a forced air circulation is used. but the circulation speed of the air around the testing sample can not be more than 1. 7 m/ s in order to prevent the appearance of the unrealistic temperature conduction inside the testing sample. The temperature around the testing sample must be within ± Z'C of the desired testing temperature and the temperature gradient does not exceed 1 'C / m. Because of the changes of the environment temperature, the conventional measurement methods and devices can't be used. For example, the torque measurement instrument for USM performance operates in temperature of 0-5S'C. Therefore, the measuring device must be installed outside the test chamber, and connected to the output shaft of USM by using a coupling shaft through a holc in thc wall of the test chamber. Thus, thc torque and speed of thc USM can be measured outside the temperature chamber. Beforc the tcmperature environment testing, at first, the surfacc quality of a ncw samplc USM has to be chcckcd. Thcre must bc no corrosion, injury, scratch and coating layer spalling on the USM surfacc. Fastcners must be firmly connected, and the connect cable must be corrcct and intact. Then, thc USM's load characteristics arc measurcd. Figurcs 14.18 and 14. 19 show the mcchanical charactcristics curvcs of TRUM-60 in high/low tempcrature environmcnt, respectively. It can bc sccn from the two figurcs that the speed and torque of the USM
432
Ultrasonic Motors Technologies and Ap plicalions
140
~
120 100 .S
i .,
0. Vl
80 +25 'C(Bcforc Icst)
- 0-
60
- o- -O·C
40
- v- - 25 'C - w- - 40 'C - _ +25 'C(Aflcr Icst)
20 0
v
- 20 - 0. 1
0.0
0. 1
0,2
0 ,3
0.4
0,5
0.7
0.6
0,8
Output torque/(N' m)
Fig. 14. 18 Measured mechanical characteristics for TRUM-60 under low temperature environment 140 120
~
~
""~
100
:So S
~'"
'"0. Vl
80 60
-A-
+25 'CCBefore test) +50'C
40
-0-
20
-1>.- +70'C -.- +80'C - 0 - +25 'CCAfter test)
0
-*- +60'C
-20 0,0
0,2
0.4
0,6
0,8
1.0
Output torque/(N' m)
Fig. 14. 19 Measured mechanical characteristics for TRUM-60 under high temperature environment
become smaller under low temperature environment. In high temperature environment, the torque increases while speed becomes lower. After finishing tests in the high/ low temperature, we measured again its mechanical characteristics at room temperature. Compared with the results before the high/ low testing, the speed and torque at room temperature changes little.
14.5.2
Vacuum Environment Testing
According to thc rcquircmcnts of USMs uscd for spacc aircraft and scmiconductor preparation, it is necessary to study its load characteristics under vacuum condition. Specifically, we need to develop testing methods in vacuum environmcnt, cstablish relatcd tcsting cquipmcnts, cxplorc thc load charactcristics of USMs as function of thc vacuum lcvel, ctc. In thc study, PDLab dcvelopcd a set of test devices consisting of an optical encoder, hysteretic machine, torque sensor and flexible coupling, as shown in Figs. 14. 20 and 14. 21.
Chapter 11
Fig. 14. 20
Fig. 14. 21
Testing Techniques for Ultrasonic Motors
433
Vacuum environment testing system for USM
Block diagram of vacuum environment testing system for USM
When the motor is operating. its speed ean be ealculated by using the number of pulse signals generated by the encoder. A load can be applied to the motor by the hysteretic machine, and can be changed from 0 to the stall torque of the motor by varying the current imposed on the hysteretic machine. The value of the load can be measured by the torque sensor, which is selected as a strain type since it is not sensitive to the vacuum environment. The torque sensor can convert the measured torque signal to frequency one. The signals obtained by the torque sensor and optical encoder are transmitted to the external apparatuses by binding posts inside the vacuum chamber. Using the three timing-counter of an A T89S52 type micro controller , we can measure and process the pulse signals from the optical encoder and torque sensor simultaneously, and the results are displayed in real-time. Because the signals are contaminated by interference noise in the process of transmission. the signal has to be filtered before they are sent to the microcontroller. The driver is placed outside the vacuum chamber and the driving current can be sent to the motor by the binding posts, so there is no need to consider the influence of the vacuum environment on the driver. In the testing, the load is gradually increased until the motor is stalled. The
Ultrasonic Motors Technologies and Ap plicalions
434
mechanical characteristics of the TRUM-60 under the normal and vacuum envIronment arc shown in Fig. 14. 22. respectively. from which we can sec that the mechanical characteristics of the TRUM-60 changes little under low vacuum 0010- 3 Pa) environment.
Atmosphere -.- 10 Pa __ 10-3 Pa
100
~
i
OJ OJ
75
50
<'<
Vl
25
o
o
0.4
0.2
0.6
0.8
Output torque/eN ·m)
Fig. 14. 22
Mechanical characteristics for TRUM-60 under normal and vacuum environment
14. 5. 3
Load Characteristics of USMs in Vibration Environment
In order to investigate vibration resistance capacity of USM during transportation and operating. the vibration evaluation testing is very important. At present. the vibration environment test for the USM is still m the initial stage: 36 37J. Referring to the standards for electromagnetic motors. the vibration condition of the environment test of the USM is determined as a broadband random vibration. Test grades can be classified as various levels in sequence from A to K. which denote the gradually increasing acceleration spectral density and total root mean square acceleration values. To evaluate the load characteristics of the USM in vibration environment. we must take into account the effect of the fixture of the USM. and its connecting shaft with the torque sensor on the measured results. To reduce the effect of the mechanical vibration on the measuring performance of the whole measuring device. an aluminum corrugated pipe is used as the connecting shaft between the torgue sensor and the USM output shaft. The connecting shaft can weaken the mechanical vibration transmitted to the torque sensor. At the same time. the USM must be fixed to a vibration table by a special clamp. as shown in Fig. 11. 23. During testing. the hysteretic machine imposes the load acting on the motor. the torque sensor measures the output torque of the motor. and the optical encoder measures the speed of the motor. All data can be collected and displayed in real-time by using the micro controller and its auxiliary circuit. The TRUM has a radial symmetry on the structure. so in actual testing it can be con-
Chaptcr 11
Tcsting Tcchniqucs for Ultrasonic Motors
435
sidered that the vibration environment test can be completed only through the horizontal and vertical vibration tablc with the samc clamping fixturc.
Fig. 14.23
Block diagram for vibration tcsting systcm
In order to verify the reliability of the testing equipment, we firstly conducted a sinusoidal vibration test with fixed frequency along the radial direction. Experimental rcsults arc shown in Fig. 14. 24.
100
7S
:i ~
i
50 - - Normal conditiom - - - - 4OOHz,4g _._._ .. 200Hz, 4g ........ SOHz,4g
r,I)
2S
0
0
0.2
0.4
0.6
Output torquel(N'm)
Mechanical characteristics for TRUM-60 at sinusoidal vibration with fixed frequency along radial direction
Fig. 14. 24
The testing proves that the motor vibration test can be completed by usmg thcse devices cOR :. The results show that apart from factors such as measuremcnt error, there are few changes in the motor mechanical characteristics. In the second step, the random vibration testing of the USM was conducted by the same dcviccs, and the vibration lcvcl was gradually incrcascd until thc whole vibration environment testing was satisfied. At first, the vibration table imposed a random excitation on the USM along its radial direction and thc mcchanical characteristics of thc motor werc measured under different vibration grades and without vibration state,
as shown in
Fig. 14. 25. It can be scen from the test rcsults that thc mechanical charactcris-
436
Ultrasonic Motors Technologies and Ap plicalions
tics of the USM are not affected significantly by the radial vibration. __ Wilhoul vibralion
100
---- 0,04 g'lHz - ' - · 0,2g'/Hz
75
.:
~
~
~ '"'"Co Vl
50
25
OutpUl lorque!CN· m)
Mechanical characteristics for TRUM-60 at different vibration grades random along radial direction
Fig. 14.25
Then, the vibration table imposed a random excitation on TRUM 60 in the axial direction, and the motor mechanical characteristics were measured under different vibration grades and the state without vibration. as shown in Fig. 14.26. It can be seen that the speed of the USM slightly decreases, and so is the torque. - - - Without vibratiou ---A-- 6Cm/ s')'lHz
80
~ S
~
'"'"Co Vl
--- 10Cm/s')'/Hz ---+--- 30Cm/s')'/Hz
60
40
20
0,1
0.2
0.3
0.4
0.5
0.6
0,7
Output torque/CN 'm)
Mechanical characteristics at different vibration grades along axial direction
Fig. 14.26
The USM relies on friction to transfer power, so there must be suitable prepressure between the stator and rotor. When the vibration is along the radial direction of the USM, the influence of the vibration on the pre-pressure is very small. so the motor mechanical characteristics change little. But when the vibration is along the axial direction, the pre-pressure will change. However. sznce the motor has a compact structure and the pre-pressure is as high as 3001'1, these small changes of the pre-pressure cannot obviously affect the mechanical characteristics. Only when the vibration level is high enough to loose the screws for
Chapter 11
Testing Techniques for Ultrasonic Motors
437
fiLing the motor shell and the pre-pressure cannot be provided, then the motor is forced to stop.
14. 5. 4
Load Characteristics of USMs under Strong Shock
W c can considcr shock as a spccial casc of vibration. It posscsscs transicnt characteristics. The peak value of shock can be high, but it quickly disappears and has lcss rcpctition timc. Thcreforc thc damagc causcd by shock is mainly a pcak damagc, and a fatiguc damagc occupics sccond. Sincc thc impact posscsscs instant characteristics, it is very difficult to measure the mechanical characteristics of the USM that is operating in an impact enviroment. We can measure that of thc motor only in thc non-opcrating statc. Thcn wc comparc thc mcasurcd rcsults before and after the motor test in the impact environment. In strong shock testing, the testing items of the motor include appearance check, mechanical charactcristics mcasurcmcnt and clcctrical paramctcrs chcck, ctc. According to the standards on the shock test, the shock impulse is chosen as a half-sine wave. The steps of shock test are set as follows: at first, check the initial pcrformancc of thc motor undcr thc normal condition, thcn fix thc motor on thc shock platform with a fixturc, and thc half-sinc wavc impulsc is applicd to thc shock platform. Thc standard shock tcst strcngth is shown in tablc 14. 1. Table 14. 1 Test condition
Acceleration/(m/s
2)
Shock test standards
:'-Iominal pulse duration/ms
Wave
Specd/ (m/ s)
11
Half-sine
3.11
A
500
B
750
Half-sine
2. 80
c
1 000
Half-sine
3. 75
As TRUM has a radial symmetry on the structure, the impact impulse can be imposed on the motor by the shock platform in the horizontal and vertical directions under the same clamping fixture. Finally, we conduct the motor performance test under normal state. The impact test system of the USM is shown in Fig. 14. 27. Thc USM mcchanical charactcristics beforc and aftcr thc impact tcsting arc shown in Fig. 14.28. Wc can scc that thc load charactcristics cithcr bcforc or aftcr thc impact tcsting rcsults in thc radial dircction do not changc significantly. As thc constraint of thc instrumcnts and cquipmcnt uscd in thc tcst, thc currcnt grading pcak accclcration of thc impact tcsting can rcach 100g whilc thc shock load can satisfy the requirements for standard testing spectrum type and tolerance. However, the mechanical characteristics of the motor under the impact environment are close to normal one. It may be concluded that the USM can adapt for the impact environment very well and its performance is very stable.
Ultrasonic Motors Technologies and Ap plicalions
438
100.------------------------------. - - Normal environment testing before shock
80
] ~
- --
Normal environment testing after shock
60 40
'"'"
~ 20
o -20L-~_~
o
0.1
0.2
__- L_ __ L_ _~_ _~L__ _~ 0.3 0.4 0.5 0.6 0.7
Output torque/eN ·m)
Shock testing platform and its testing system
Fig. 14. 27
14. 5. 5
Shock testing results for TRUM-60
Fig. 14. 28
Test and Analysis of Noise from Ultrasonic Motors
Though the operating frequency of USM is above 20kHz. the unpleasing nOIse (maybe above 60dB) exists in the USM due to some faetors, such as structure design, manufacturing technique, functional materials and contact interface c09 : , etc. It is important to reduce the noise to a certain extend and avoid its pollution. So it is necessary to conduct the noise test and to analyze the noise resource of the USM. Fig. 14. 29 shows the measurement system for noise.
Anechoic
USM
chamber
Sound
meier
SpeClnlln -7""-==--4...-:;11.-analyzer
Fig. 14.29
:'\raise measurement for TRUM
According to the proposed standard by Shinsei Corporation, the sound-level meter for the noise data acquisition is put on the place 100mm far from the shell of USM. We have found that the noise is influenced mainly by the driving frequency, pre-pressure, friction materials and manufacture assembly quality. We have measured the noise of TRUM-60, whose results are presented and analyzed as follows:
1. Influence of driving frequency on the noise With the driving frequencies 42. 5kHz and 39kHz (resonance points), the corresponding experimental results are shown in Fig. 14. 30, which are measured under the same pre-pressure of 250N. It is found that the higher noise is produced with the decreasing driving frequency and increasing the rotary speed of the USM.
Chaptcr 11 0.015
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Tim s (a) Acoustic intcnsity 43dB(42 .5 kH z)
-40 Time!s (b) Acoustic intcnsily 92 dB (ncar to re onance frequency 39kHz)
Fig. 14.30 :'\raise measurement results for TRUM-60 at two different driving frequencies
In fact, the piezoeleetric material is depicted by the nonlinear constitution equation under strong electric field. The stator's response will become more complex due to the nonlinear effect of the piezoelectric material, which can obtain the sharp noise of the USM. The experiments show that noise will be greatly decreased as long as the driving frequency is a little higher than the resonance frequency. So the driving frequency should keep away from the resonance frequency.
2. Influence of pre-pressure on noise Fixing the driving frequency of 35kHz and gradually increasing pre-pressure, we have conducted a series of experiments in which the results corresponding to the pre-pressure of 20)J and 210N are illustrated in Figs. 14. 31(a) and (b). respectively. It is found that small pre-pressure will produce quite louder noise. For example, the pre-pressure of 20)J results in 86dB. However with the increasing pre-pressure the noise of high frequency vanishes. The noise without the high frequency components decreases to 53dB at the pre-pressure of 210N. The experiments illustrate that the noise of the ultrasonic motor decreases remarkably with the increasing pre-pressure. It can also prove that the high prepressure can improve the contact characteristics between the stator and rotor and enlarge the structure stiffness. 3. Influence of friction material on noise Due to the application of macromolecule friction material, the noise of USM can be reduced to below 30dB. Moreover, the smooth finish of friction material has an effect on not only the life of the USM, but also its noise. Usually the friction material with the coarse finish results in the noise with low frequencies; on the other hand the noise with high frequencies is produced by the good smooth finish, which will reduce the noise level. 4. Influence of manufacture and assembly quality on noise Manufacture and assembly errors lead to the contact nonuniformity between
Ultrasonic Motors Technologies and Ap plicalions
440
0. 15 r-~------~--~--.., 0. 1
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~
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Noise measurement results of ultrasonic motor with various pre-pressure
the stator and rotor, then the impure traveling wave appears in the stator. In this case the noise of the USM can be produced. So we have to improve the smooth finish, reduce assembly errors, and adj ust appropriate pre-pressure, so as to decrease the noise. Then TRUM-60 and TRUM-10 developed by PDLab were chosen as noise testing samples, which were placed in anechoic room of Aerodynamic Institute of )JUAA. The experimental results show that the noise level of TRUM-60 and TRUM-40 are 38. 5dB and 29dB. respectively.
14. 5. 6
Testing of USMs in Hygrothermal Environment
The experiments are proposed to research on the adaptability of USM simulated hygrothermal environment. The key of the hygrothermal test is how to simulate the natural humidity of the envionment under a shortening testing period. Considering the testing features, the moisture absorption characteristics of friction materials will be measured besides the mechanical characteristics of the USM. Some special methods can be applied to shorten the testing period. such as increasing temperature and enhancing relative humidity. Fig. 11. 32 illustrates the alternation of the temperature and humidity in the hygrothermal testing. 95 -.-.-.-.-.~ ._._._._._._._._._._._. _._._._._._.- 95 -·_·-Humidity -: 85 85 ---------Temperature : :
55
I I:
30
, - - - - i..
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1. Test chamber and project In the hygrothermal experiment, the available room should accord with the testing rules and allowable errors. The water used in the experiment must be purified. The condensation water on the top of the testing chamber should fall on the tested samples drop by drop, so as to avoid the seepcr on the samples. Of course. this is very distinct from the natural hygrothcrmal environment. The high/low temperature hygrothermal test chamber from ACS Co. of Italy is applied and the available temperature ranges from -75"C to 180"C, and humidity from 10 percent to 98 percent. The experiments inelude the mechanical characteristics of the USM before and after the hygrothermal environment and the moisture absorption characteristics of friction materials. 2. Mechanical characteristics of the ultrasonic motor in the hygrothermal environment Figure 14. 33 depicts the mechanical characteristics of the USM in different hygrothermal environments. We notice that a normal USM is first measured before the hygrothcrmal environment. then once again after 24h storage in the hygrothermal environment. Moreover the USM is measured after 18h storage in the hygrothermal environment, and then measured once again after the hygrothcrmal environment. It is found that the USM can operate well in the hygrothcrmal environment. 3. Moisture absorption characteristics testing of friction materials Due to the contact mechanism of USM. the moisture absorption of friction materials with a long time storage can lead to the adhesion between the stator and rotor. In this case the USM's startup is more difficult and so the moisture absorption characteristics testing of friction material is very necessary. By weighing the tested samples before and after the storage in humid environment. we measured and calculated the moisture absorption characteristics of three kinds of friction materials, respectively. based on poly(cthcr-cther-ketone) (denoted by No. 1), 150
;:R ~
125 ::
100
~
75
"'-
50
~
g
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.~
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.0
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;;; 0.2 <;
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::E 0.1 0 OlllplillOrqllel(N ' m)
Fig. 14. 33 Mechanical characteristics for TRUM-60 in normal and hygrothermal environments
Hygrol hermal
Hygrolhermal
Hygrothermal
Moisture absorption ratio for three kinds of friction materials
Fig. 14.34
442
Ultrasonic Motors Technologies and Ap plicalions
polytetrafluorethylene (denoted by :'\10. 2 and :'\10. 3 with different formulas). Fig. 14. 34 illustrates that the friction material based on peek has high moisture absorption ratio. which is of one order of magnitude higher than that based on polytetrafluorethylene. It is found that friction material based on polytetrafluorethylene with the certain formula (No.2) is adapted to the USM with the storage for a long time in the hygrothermal environment.
14.6
Life Testing for USMs
In order to industrialize the USM. its endurance performance has to be evaluated. Some countries have adopted the variety of abrasion-resistant materials. to coat the surface of the rotor or stator. or to directly conduct the surface treatment on the wear surfaces of the rotor/stator. so the life of one ultrasonic motor has reached more than 3 OOOh. PDLab developed a fatigue test system consisting of a computer. peripheral devices and corresponding software. and successfully conducted the fatigue testing for traveling wave rotary USMs: 40 42J. The test system can be used for motor testing under load or non-load conditions. and the service lives of the motors can be obtained. respectively. USM service life depends on the failure time of its vulnerable components. The major vulnerable components of the USM are the piezoelectric ceramic ring and the friction material layer. The two failure reasons of the piezoelectric ceramic ring are shown as below: (1) the piezoelectric ceramic material itself is defective or the bonding layer between the piezoelectric ceramic ring and the stator fails. which can lead to the piezoelectric ceramic ring's rupture due to stress concentration or its shedding due to weak bonding; (2) the operating frequency is elose to a natural frequency of the stator and the amplitude of stator is larger. which can make the piezoelectric ceramic ring to rupture. It can be seen that the failure of the piezoelectric ceramic ring is abnormal failure. It can be avoided by choosing high-quality piezoelectric ceramic ring. standardizing bonding process and selecting suitable operating frequency. The failure of the friction material layer is inherent because of the friction driving between the stator and rotor. Under normal circumstances. the failure time of the piezoelectric ceramic ring can be far more than that of the friction material layer. So the USM's service life mainly depends on the failure time of the friction material layer. Here. the cumulative running time and total running cyeles of a USM are used as indicating the USM life.
14. 6. 1
Design of Life Testing System
In order to increase the life of USM. the stator (or rotor) contact surface is coated with an abrasion-resistant material. :'\Ievertheless. the current USM's life
Chaptcr 11
Tcsting Tcchniqucs for Ultrasonic Motors
443
can only achieve 5 OOOh at most. Moreover, friction produces heat and increases the temperature of the motor, and excessive temperature rising damages bonding agents and the piezoelectric ceramic ring. Due to these factors, USM can only operate in intermittent and low speed (from dozens to hundreds rounds per minute) conditions. In order to simulate the intermittent operating state of USM at testing, the motor stops for a period of time after the motor has run a certain number of rounds, then it forms such a cyele. The measured data have two major items accumulated running time and the rounds of the USM. Therefore, the testing system should have the following features: (1) a programmable start-stop function can make the system stop the motor rotation when the motor turns certain rounds and restart it when the prescribed stop time reaches; (2) an automatic recording system can record operating rounds and time and calculate the average speed; (3) automatic shut-of[ function, can make the motor shut down automatically while the system does not receive the signals of operating in the prescribed time. The parallel interface of the computer can be used for completing the above functions. The latch in the parallel interface can remain a high or low level. It is often used for the simple data acquisition and control systems with fewer I/O ports and low-sampling speed, which can transfer data immediately without specially equipping interface circuit, and a number of input and output pins can be selected. According to the characteristics of USM life testing, data are transmitted by parallel interface, which makes the system simple and connections convenient to satisfy the testing requirements. The main functions of the testing system are able to run the motor intermittently and record the accumulated running time and rounds. Therefore, the hardware of the system should have the following functions: (1) it converts USM running time and rounds into electrical signals and sends them to the computer; (2) the output control signals from the computer are amplified and used for controlling the motor to run intermittently. Based on the above requirements, we designed the hardware of the life test system, of which block diagram is shown in Fig. 14.35. It can be seen that the hardware circuit of the test system mainly consists of a detection circuit for calculating operating rounds, and an amplification circuit for detecting and controlling signals. The detection circuit is composed of a photoelectric switch and a shading plate with gap. The shading plate, which is fixed on the motor shaft, is an opaque circular disk with a narrow slot in its radial direction. When the motor turns a round, light from a light-emitting diode (LED) can be sensed by a phototransistor through the narrow slot and make the circuit output a pulse signal with high electric level. Thus, the motor rounds can be converted into the pulse signal. The number of the pulse with high electric level is rounds of the motor. The time interval between pulses with high electric level is the motor running time.
444
Ultrasonic Motors Technologies and Ap plicalions
Hysteretic
Data acq ui sition card (a)
Signal analysis and control system
USM operation
rounds detection
Parallel interface
of computer (data acquisitipn and control software)
US M drive (b)
Fig. 14. 35
14. 6. 2
TRUM We testing system
Life Testing Results and Analysis for TRUM
The purpose of the endurance test of USM is to extend its life. In this way the motor can meet the requirements of the engineering applications. Since the USM life depends mainly on the used friction materials, the purpose of the endurance test is to seek suitable friction materials for the USM. PDLab has conducted endurance testing of USMs, which are bonded with various friction materials and obtained the life data of the friction materials, the changes of the motor speed in operating and the phenomena presented during the testing. During the initial stage of the test, the driver of the motor could not automatically track the frequency change as running time increased and the interface wear increased, so the motor operating state was severcly affected and its speed changed greatly with time. Moreover, the forming abrasive dusts between the stator and rotor easily sticked on the contact surface and formed larger particles. When the particles became big enough, the effective contact area between the stator and rotor was greatly decreased such that the motor slowed down or even stopped. Based on the analysis of the endurance testing results, phenomena and data, some improved suggestions were obtained for the friction layer C<3 " : . First, wear resistance of friction pairs must be high, that is to say, in the same conditions and within the same time the abrasive dusts are produced as little as possible. Second, the hardness of friction pair must be high and its viscosity must be low, and the sizes of the abrasive dusts should be as small as possible such that the dusts can be easily removed out of the friction interface. Through the improvement of friction materials, the no-load service life of the TRUM-40 was more than 5 80oh. Its speed and time change curves arc shown in Fig. 11. 36. The life testing of the service life of TRUM-60 under the cycle load was also measured (30min empty, 30min half rated load, 30min rated load, then
Testing Techniques lor Ultrasonic Motors
Chapter 11
445
30min no-load, where the rated load is O. 5N-m). 'lotal running time was more than 310oh. Their speeds vs. time eurves are shown in Fig. 14.37. 160 140 120 -§ 100 80 860 ell 40 20 0
100.---------------, 90 80 70
i
.§" 60 50 ~ 40
:s.
• Dala poilll
- Fittin g curve
0
100
200
300
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400
500
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Fig. 14. 36 :'-la-load We testing for TRUM-40
• Data poinls
30
- Filling curve
20 10
o+-----~----~----~----~~
o
500
1000
1500
2000
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References [ 1 JAM Flynn. Performance of ultrasonic mini-motors using design of experiments. Smart Material Structures, 1998, 7(3): 286-291. [2 J
A Endo, N Sasaki. Investigation of friction material for ultrasonic motor. Japanese Journal of Applied Physics, 1987, 26(1): 197-199.
[ 3J [ 4J [ 5J [ 6J [ 7J
B Armstrong. Friction: experimental determination, modeling and compensation. IEEE 1998 IeRA, 1998, 3: 1122-1127. T Morita, S Takahashi, H Asama, et a1. Rotational feed through using an ultrasonic motor and its performance in ultra high vacuum conditions. Vacuum, 2003, 65: 53-57. T Morita, T Niino, H Asama. Rotational feedthrough using ultrasonic motor for high vacuum condition. Vacuum, 2002, 65: 85-90. I Takaaki, M Eiiehi, N Kentro, et a1. Characteristics of ultrasonic motor driven in a vacuum. Japanese Journal of Applied Physics, 1998, 37: 2956-2959. M F Six, R L Letty, R Seiler, et a1. Rotating piezoelectric motors for high precision positioning
[ 8J
[ 9J [lOJ
[llJ
& space applications. 9 th International Conference on .1'Vew Actuators. Bremen, Germa-
ny, 2004: 714-717. )/a Suo Experiment Study on Ultrasonic Motor in Abnormal Environment. Dissertation for the Degree of Master. )/anjing: Nanjing University of Aeronautics and Astronautics, 2007. (in Chinese) Qing He, Wolfgang Seemann. Modal analysis of a ring type linear ultrasonic traveling wave motor. Piezoelectric and Acoustooptics, 2003, 25(6): 511-516. (in Chinese) Na Su, Dan Lu, Chunsheng Zhao. Experimental research on load characteristics of ultrasonic motor under vacuum condition. Journal of Vibration, Measurement and Diagnosis, 2006, 26(S2): 151-153. (in Chinese) Yonggeng Lu, Chunsheng Zhao. Research and manufacture of pre-pressure regulate apparatus [or ultrasonic motor based on single chip microprocessor. Piezoelectric and Acoustooptics, 2006, 28(1): 51-52. (in Chinese)
[l2J [13J [14J
Yuebo Ji, Chunsheng Zhao, Shuren Qin. Exciting signal for ultrasonic motor testing basing on personal computer. Journal of Chongqing University, 2006, 29 (1): 12-14. (in Chinese) Chunsheng Zhao. Some proposals for development of ultrasonic motor techniques in China. Micromotors Servo Technique, 2006, 39(2): 64-67. (in Chinese) Cunyue Lu, Jie Li, Chunsheng Zhao, et a1. Experimental tests of longitudinaltorsional ultra-
446
[l5J [l6J [l7J [l8J
[l9J
[20J [21J
[22J [23J [21J
[25J
[26J
[27J [28J
[29J
[30J
[31J
Ultrasonic Motors Technologies and Ap plicalions sonic motors. journal of Tsinghua University (Science and Technology) , 2006, 46 (6): 851851. (in Chinese) Azhou Zhang, Kerong Zhang, Qihang Yao, et at. Vibration Environment Engineer. Beijing: Aviation Industry Press, 1986. (in Chinese) Chuanrong Zhou, Chunsheng Zhao. Parameter Identification and Its Application of Mechanical Vibration. Beijing: Science Press, 1989. (in Chinese) Chunsheng Zhao, Weiqing Huang. Experimental study on ultrasonic motors (USMs). journal of Vibration, Measurement and Diagnosis, 2003, 23(1): 1-5. (in Chinese) Zhenhua Xiong. Study on Dynamic Characteristics of Stator of Piezoelectric Ultrasonic Motor. Dissertation for the Degree of Master. Nanjing: Nanjing University of Aeronautics and Astronautics, 1998. (in Chinese) Zhenhua Xiong, Chunsheng Zhao, Qidong Chen. Study on modal characteristics of stator of flex-torsion coupling ultrasonic motor based on standing wave. The 8th Modal Analysis and Experiment Conference in China. Shanxi, 1998: 98-103. (in Chinese) http://www.Polytec.com Hui Guo. Study on Ultrasonic Motor Based on In-plane Vibration Mode. Dissertation for the Degree of Doctor of Philosophy. :'-Ianjing: :'-Ianjing University of Aeronautics and Astronautics, 2001. (in Chinese) Yuebo Ji, Chunsheng Zhao. Test system for ultrasonic motors in-plane vibration modal testing. Chinese journal of Mechanical Engineering, 2006, 12(2): 110-111. (in Chinese) Guoguang Yang. Measurement Technology of Modern Optics. Hangzhou: Zhejiang University Press, 1997. (in Chinese) Yongming Bai, Yuebo Ji, Weiqing Huang. Test system for ultrasonic motors in-plane vibration based on LDV. The 11th China Small Motor Technology Conference. Shanghai: No. 21 Research Institute of CETC, 2006: 182-183. (in Chinese) Yuebo Ji. Measurement System for Ultrasonic Motors In-plane Vibration Modal Testing. Post-doctoral Report. Nanjing: :'-Ianjing University of Aeronautics and Astronautics, 2006. (in Chinese) Yongming Bai. Study on Measurement System for In-plane Vibration Modal. Dissertation for the Degree of Master. :'-Ianjing: Nanjing University of Aeronautics and Astronautics, 2007. (in Chinese) Yonggeng Lu, Chunsheng Zhao. Program design for pre-pressure regulate apparatus of USM. Machinery and Electronics, 2004( 10): 38-40. (in Chinese) Qingqing Han. Experiment Study on Pre-pressure of Ultrasonic Motor. Dissertation for the Degree of Master. Nanjing: :'-Ianjing University of Aeronautics and Astronautics, 2006. (in Chinese) Ming Yang, Chunsheng Zhao. The property test of annular traveling wave ultrasonic motor by measuring step response. Piezoelectric and Acuustooptics, 1998, 20 (5): 347-394. (in Chinese) Shoulin Shen, Chunsheng Zhao, Weiqing Huang. Wavelet transform applied to testing and analysis on instantaneous characteristic of ultrasonic motor. IEEE International Ultrasonic Symposium, 2003, 2: 1778-1781. Chunsheng Zhao, Shoulin Shen. Test and analysis of starting-up and stopping response characteristics of ultrasonic motor using wavelet transformation. ] uurnal oj Data Acquistitiun
and Processing, 2004,19(4): 376-380. (in Chinese)
[32J [33J
Ming Yang, Chunsheng Zhao. The simulation and test of the annular traveling wave ultrasonic motor. Acta Acustica, 1999, 24(5): 484-490. (in Chinese) Heling Suo Mathematical Mode and Simulation of Standing Wave Type Ultrasonic Motor Driven by Single Phase Singal. Dissertation for the Degree of Master. :'-Ianjing: Nanjing
University of Aeronautics and Astronautics, 2001. (in Chinese)
[34J
Shoulin Shen, Chunsheng Zhao. Starting characteristic testing and analysis of ultrasonic motor based on wavelet transformation. Wavelet Analysis and Its Applications (WAA). Chongqing, China, 2003: 911-119.
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[38J
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Shoulin Shen. Study on Measurement Technology of Ultrasonic Motor. Post-doctoral Report. Nanjing: Nanjing University of Aeronautics and Astronautics, 2001. (in Chinese) Aiwen Gu, Weiqing Huang. Experiment of ultrasonic motor in vibration environment. ] ournal of Vibration Engineering, 18(S): 141-144. (in Chinese) Aiwen Gu. Experiment Study of Ultrasonic Motor in Vibration Environment. Dissertation for the Degree of Master. )/anjing: Nanjing University of Aeronautics and Astronautics, 2006. (in Chinese) Dan Lu, )/a Suo Experiment study of ultrasonic motor in vibration environment. The 11th China Small Motor Technology Conference. Shanghai: )/0. 21 Research Institute of CETC,
2006: 175-177. (in Chinese) [39J
Dan Lu. Study on Environmental E.rperiments for Traveling Wave Type Rotary Ultrasonic motor. )/anjing: A dissertation for the Degree of PhD of Nanjing University of Aeronautics and Astronautics, 2008.
[40 J
Qianwei Chen, Weiqing Huang, Chunsheng Zhao. Measurement of service life of ultrasonic motors (USM). ] ournal of Vibration, Measurement and Diagnosis, 2001, 21 (l): 19-22. (in Chinese)
[41J
Qianwei Chen. Study on E.rperiment Technologies of Ultrasonic Motor. Dissertation for the Degree of Master. Nanjing: Nanjing University of Aeronautics and Astronautics, 2004. Wei Zheng, Chunsheng Zhao. Study on life experiment of traveling wave type ultrasonic motor. The 11th China Small Motor Technology Conference. Shanghai: No. 21 Research Institute of CETC, 2006: 171-174. (in Chinese)
[ 12J
[43J
[44J
Wei Zheng, Chunsheng Zhao. Study on wear mechanism of traveling wave type ultrasonic motor. The 11th China Small Motor Technology Conference. Shanghai: No. 21 Research Institute of CETC, 2006: 136-110. (in Chinese) lianjun Qu. Study on Contact Model and Friction Material Characteristics of Ultrasonic Motor. Post-doctoral Report. Nanjing: Nanjing University of Aeronautics and Astronautics,
2001. (in Chinese)
Chapter 15
Applications of Ultrasonic Motors in Engineering In the early 1990's. USM started to enter the practical application stage via representative manufactures such as Shinsei Co., Ltd. and Canon Co., Ltd. In addition. many countries including the USA, Germany. France, Britain. Israel, Denmark, etc. put in a lot of manpower and resources to develop ultrasonic motors, and now there are many new entrants in the market, such as PI (Physik Instruments) L. P. in Germany. DTI, LLLP, and New Scale Technologies, Inc. in USA. Nanomotion Ltd. in Israel, PCBMotor ApS in Denmark, etc. Research on and applications of USMs are being gradually promoted in China. At present, there are over 30 universities and research institutes conducting research in this area, some of which can produce a small batch of USMs. )JUAA. Tsinghua University, Zhejiang University, Jilin University, etc. have achieved fruitful results. According to e.Lperts' estimates, with further research and the wider applications of USMs, there will be an ultrasonic motor market worth tens of billions of dollars in the world. If the applications of USMs in the automobile and office automation equipment industry arc fully developed, the market prospect will be incalculable- 1-. Fig. 15. 1 shows the examples of USMs' engineering applications. USM
USM
USM
(a) DVD pick up syslem
(b) : .., is siage
(c) Semic.onductor equipmenl
USM
USM
USM (d) Micro moving system (e) Diamond CUlling machine
Fig. 15. 1
(I) Roller screen
(g) Chassis posilion syslem
Applications of USMs in general engineering
C. Zhao, Ultrasonic Motors © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011
Chapter 15
15. 1 15. 1. 1
Applications of Ultrasonic Motors in Engineering
449
Applications in Domestic Engineering Application in Camera
Professional photographers know that in order to obtain good autofoeus effect. the movement of a lens actuation device must be quick, quiet and accurate. However, traditional autofoeusing lenses produce operating noise similar to that induced by the gear of robot joints, and the focusing process is slow. In comparison, USMs are able to meet these requirements. Canon Co., Ltd. became the first camera maker to apply USMs in the autofoeus systems of the EOF series of cameras in 1987. Since the EF 300mm £/2. 8L lens with the USM was introduced into the market, the camera industry has been amazed by its silent and super-fast autofoeus performance. In 1990, Canon Co., Ltd. developed a ring-type USM shown in Fig. 15. 2, which can be used for a variety of more affordable lenses. The velocity of the USM is 37r/min and the torque is O. 03)J·m L2J • Fig. 15. 3 shows the location of the USM in the camera.
Fig. 15. 2
Ring-type USM
Fig. 15. 3
Ring-type USM in camera
Features of the ring-type USM inelude its ability to easily achieve the lowspeed, high torque for direct drive. A large holding torque means that the disc brake automatically holds the lens in the place where the motor is stopped. The construction of the USM is simple, its operating is virtually noise-free, and it demonstrates excellent start/stop response and controllability. High efficiency and low power consumption allow the lens to be powered by the camera's battery. The motor's ring shape is optimally suited to lens barrel applications and its low rotation speed is ideal for lens drive purposes. The rotation speed covers a wide variable range from o. 2r/min to 80r/min to realize high precision, high speed drive and control of lens. The electronic manual focus system with variable sensitivity is also available. The broad operating temperature range of - 30"C to +60"C (-22"F to 140"F) ensures stable operating even in severe environments. Moreover. all lens drive and control are performed by the microprocessor housed within the lens. Fig. 15.4 shows the block diagram of the control system L3 -. The frequency automatic tracking is realized by adjusting the phase shift between electrode S and electrode A. The control signal is given by the CPU using the signal from the rotary encoder, which is directly connected to USM.
Ultrasonic Motors Technologies and Ap plicalions
450
USM stator
AMP
m
Output port
outr---,-------------------,-~
programmable oscillator CW/CCW
Output port
mtput port(l)
90"
phase shift out
Enable
f-------------~P~h~a~se~c~o~m;p~ar~ato;;;;-rI---;:::;:=:::;:::;:::~ Signal of resonance
CPU mtput port(2) r---------,--:------1 Level converter Detect position L-__________--' and speed
Photo transistor _
·0 1
mtput port(3)F===~~m~t~ern~a~c~e~lo~g~ic~&~UP~ID~O~WN~~c~o~un~t~erj====::::J
Fig. 15. 4
_
Optical encoder (connected to USM rotor) LED
Manual ring Precision rotary encorder
Block diagram of drive and control system for lens-focusing using USM
The feat was followed in 1992 by the developing the new type of micro USM that enabled the automation of production, as shown in Fig. 15. 5. Its location in the lens is shown in Fig. 15. 6. The velocity of the USM is 930r/min and the torque is o. 07='J om: 4 ]. Because of lower manufacturing cost. the USM is used for the middle and low cost EF lens.
Fig. 15. 5
Bar-type micro USM
Fig. 15. 6
Bar-type micro USM
In camera
Another tra vcling wave type ultrasonic motor. micro USM II (as shown in Figs. 15. 7 and 15. 8), was produced by Canon Co., Ltd. in 2003. The motor is about half the size compared with the previous micro USM. Although the weight of the stator is about one fifth of the previous micro USM, its output power is about the same. The cost is said to be about several hundred] apanese yen. while the cost of previous micro USM is about one thousand Japanese yen[5]. Stacked PZT in a body is used to reduce the driving voltage (the driving voltage of previous USM is 100V,,,,,, the new one is 12V""J and cost, and to increase the efficiency-6J. At present, the USM can be seen in standard zoom lenses, ultra-wide zoom
Chapter 15
Applications of Ultrasonic Motors in Engineering
451
Slide bearing Output ge
rrr___~it:3lr-- Rotor ring
UJ.....~--t-- Supporti ng rod .41----1._-
Flexible rotor
......--:c!:o-- Friclion plate t-""'=i=::!....- Stacked PZT Screw Bouom plate
Fig. 15.7 Traveling wave type ultrasonic motor, Canon micro USMII
Fig. 15. 8 Sectional view of Canon micro USMII
lenses and telephoto zoom lenses produced by Canon Co., Ltd.
15. 1. 2
Application in Cell Phone
With the continuous improvement of people's standard of living, additional requirements for cell phones have become increasingly diversified. Among them, the cell phone's camera is favored by the fashionable young people. Traditional electromagnetic motors are hardly used for the optical zoom of the cell phone's camera, because the size of the cell phone itself is very small, even smaller size of the motor is required. But when the size of the traditional electromagnetic motor is reduced to a certain extent, its output torque and efficiency will decline sharply. Therefore, the previous cell phone systems just usc a digital camera zoom. While the digital zoom reduces the sharpness of the image, so it is optical zoom that can solve this problem better. At present, South Korea's Samsung Corporation has applied two USMs (5mm long and diameter of 1. 6mm) in each cell phone. Due to the minimal increase in the size and weight of a cell phone, a set of optical zoom is installed and 4 times optical zoom can be achieved.
15. 1. 3
Application in Watch
As early as in 1991, Seiko Instruments Incorporation developed a traveling wave type ultrasonic micro motor with the diameter of 10mm and the thickness of 5mm and used for a commercially available watch. However, this motor has some practical problems in use, i. e. the driving circuit and structure of the motor are very complicated and difficult to miniaturize further. To overcome this problem, researchers have set their sights on the typical quartz oscillator, which constructs a self-oscillating circuit as a driving circuit. On October 1996, Seiko Instruments Incorporation succeeded in applying the ultrasonic micro-motor (8mm diameter) to the vibrating alarm watch, as shown in Fig. 15. 9. Fig. 15. 10 shows an overall view of the vibration alarm. The off-center weight is placed on the outside of the projections on the vibrator. Tungsten, which has a large specific density, is used as the weight to reduce a centrifugal force, which is the source
Ultrasonic Motors Technologies and Ap plicalions
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of the vibration L7J •
Projections
Fig. 15. 9 Ultrasonic micro motor used for vibration alarm
Stator
Fig. 15. 10 Overall view and expanded view of vibration motor
Until now, to create a perpetual calendar watch using conventional technique, an additional stepping motor and many reduction gears are required to obtain the high torque, and to move the date wheel for two or more dates in the case of months with 30 days (or less). This would make the configuration of the calendar mechanism complicated and the size of the movement larger. To overcome this problem, researchers used an ultrasonic micro motor as a driving source for the calendar mechanism, as shown in Fig. 15. 11. Fig. 15. 12 shows the appearance of the ultrasonic micro-motor used in this mechanism. This ultrasonic micromotor is 4. 5mm in diameter and 2. 5mm in thickness. The characteristics of this ultrasonic micro-motor arc that the starting torque is O. 02mN'm and the noload rotational speed is 2 OOOr/min at 3Vdc ' The driving frequency is about 630kHz. The starting torque is more than ten times that of the electromagnetic stepping motors used in watches. Date jumper
Date wheel
Intermediate date wheel
24-hour wheel
Date dial
Fig. 15. 11 Ultrasonic micro-motor used in calendar mechanism
15. 2 15. 2. 1
Fig. 15. 12 Calendar mechanism using ultrasonic micro-motor
Applications in Industrial Engineering Application in Gasoline Generator
Without automatic voltage regulation, the output voltage of a portable gasoline
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generator will decline because of a heavier load. The usual solution is to manually raise the power output of the gasoline engine. Obviously. it is very inconvenient and uneconomical. To make the generator to adjust automatically its output voltage. a feedback must be introduced and the mechanism for gun regulation must be added. Due to a lot of advantages of USMs, they become one of the best choices for gun regulation. Figure 15. 13 shows the block diagram of the system for automatically adjusting the gun of the gasoline generator. The system consists of the TRUM-40 ultrasonic motor made by PDLab. gasoline generator. the computer and AID. DI A. and so on. Through the AID converter interfaces. the computer samples the output voltage at the frequency of 20 Hz. After calculation, the computer gives control commands and adjusts the gun by changing the speed of the ultrasonic motor to keep the output voltage stable in the designated range. Fig. 15. 14 shows how to automatically adjust the gun of the gasoline through the USML8J.
Adjusting device for the gun of gasoline with USM
PC Fig. 15. 13 Block diagram of system for automatically adjusting gun of gasoline generator
15. 2. 2
Fig. 15. 14 Automatically adjusting gun of gasoline through USM
Application in Automobile
USMs have a very broad application in the car. In Fig. 15. 15, the small round balls are used to mark the locations of a car. which can use the ultrasonic motor.
Fig. 15. 15
Layout of applications of USMs in a car
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The aim of rearview mirrors on both sides of a car is for observing traffic condition. To park or reverse the car, the driver must adjust the mirrors in the vertical and horizontal directions to observe differcnt regions of the road. Thus semi-automatic and fully automatic rearview mirror system appeared. A direct way is to usc a motor to rotate the rcarview mirror around two coordinatc axcs. Traditional elcctromagnctic motors need gcar mechanism to reducc thc speed, but thc mcchanism occupies larger volumc, and adjusts thc speed slowly. Due to the smaller size, lighter weight, lower speed, and higher torque of USMs. in somc automobilcs madc by thc Japan. USMs havc bcen used to drivc the rearview mirror, as shown in Fig. 15. 16. Since there is no gearbox between thc ultrasonic motor and thc mirror. the output shaft of the USM can be dircctly connected with the mirror framc and the USM itsclf is fixed with thc car body. The compact USM can be integrated with the rearview mirror, which makes it elegant in appcarance and convenient for adjustmcnt. Another application of USMs in automobile is shown in Fig. 15. 17. If the angle of the headrest attached to top of the seat can be adjusted according to the needs of passengers, the occupant can feel more comfortable and their cervical can be protected. The motor for adj usting the headrest should be small in size, quict in opcrating, and capable of providing with a torquc and a sclf-Iocking torquc. Traditional motors arc cither too large in size or too small in torque to meet these requirements. The features of USMs make them meet these application conditions.
- -- ---- -- .....
"- ,
USM
r1\ I I \
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II
II
Fig. 15. 16
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USM used for driving mirror
Fig. 15. 17
USM used for headrest
In order to meet the needs of the people with different size and height, the position of an automobilc stccring wheel should be adjustable. Comparing with a convcntional DC motor uscd in a steering whcel adjustmcnt dcvicc, USMs havc a lower operating noisc level whilc maintaining a similar output torquc. Thcreforc, some vehiele manufacturers use USMs to replace the DC motors- 9- . The structure is shown in Fig. 15. 18. USMs in Fig. 15. 18 are chosen as some examples to briefly describe the application. When USMI is energized for forward rotation, the rotor of the motor rotatcs and thc rotation is transmitted to the scrcw shaft through transmission componcnts, such as gcar. The forward rotation of the scrcw shaft lcads to
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Applications of Ultrasonic Motors in Engineering
USMI
USM2
Gear
455
Steering wheel
Screw nut Screw shaft assembly
Fig. 15. 18
Steering wheel position adjusting apparatus
movement of the screw nut assembly, so that the movable tubular column is moved about the horizontal shaft in a eounter eloekwise direetion, and the steering wheel is tilted correspondingly. The tilting movement of the steering wheel position is stopped when the motor is de-energized and the tilted steering wheel position is maintained by engagement of the screw shaft with the screw nut assembly. Similarly, the steering wheel can be shifted in the rear or forward position by USM2. In order to steer the steering wheel more efficiently, there is a torque sensor under the steering wheel, as shown in Fig. 15. 19. It is used to detect the torque applied to the steering wheel by driver, and control a motor to produce an auxiliary torque. The motor is generally an electromagnetic motor and the gear reducer is required to obtain a low speed and high torque. Viewing from the steering wheel, the motor system (motor and gear mechanism) has a considerable moment of inertia, which is that of the rotor itself multiplied by the square of reduction ratio (about 10). The moment of inertia is so large to lead a slow response of control system, and to induee possibly a low resonance frequency due to the flexibility of surrounding parts. If the ultrasonic motor is used, the speed reduction mechanism can be removed, the equivalent inertia will be greatly reduced, then the mobility of the automobile is greatly improved. Steering wheel
Control circuit
Torque sensor
Gear reducer
Small gear
Fig. 15. 19
Constitution of a power steering device
456
15. 2. 3
Ultrasonic Motors Technologies and Ap plicalions
Applications in Robot
Previously, robots are usually driven by traditional electromagnetic motors. With the continuous development and improvement of USMs' technology, USMs are more and more widely used in robots. A cleaning robot driven by USMs is shown in Fig. 15. 20. The locomotion system is a differential driving deviee, i. e. the each whecl driven independently by one TRUM-40 made by PDLab, so the robot has the ability to move straight. turn in place, and to move in an arc: 10: . The robot is controlled by DSP. With touehing, infrared and photosensitive sensors, the robot ean sense its surroundings. It ean avoid obstacles, roam and reach a eertain path planning. In the process of moving, it can clean the ground by means of its own portable vacuum deviee. Just pushing a start button. the robot with power supply ean start to operate. After finishing cleaning. it will return to some light plaees and automatically stop the motion.
Wrist joint -
----
Shoulder
_ _I:-+- Gripper
__-"'--'-----,,__- Rotary
joint
encoder Wai st joint - - - - - - 1
Fig. 15.20
Vacuum cleaning robot
Fig. 15.21
Joint robot
Figure 15.21 is a robot that can draw figures and write words on any flat plane in its workspace[ll:. Three joints are all driven by USMs made by PDLab. Two TRUM-15 are used for the waist and wrist joints. respectively. and one TRUM60 is used for the shoulder joint. In order to obtain the actual position of the each joint, every joint is equipped with one incremental rotary encoder (2 OOOp/ r). Thanks to the large moment at low speed of the USMs and the hollow shaft encoders, there are no gear box in the driving devices and no connectors between the encoders and USMs, so the configuration of the robot is compact and simple. The actuating range of the wrist and shoulder joints is ± 90° and that of the waist joint is ± 360°. Figure 15. 22 shows a Seiko Epson Corporation's Monsieur II-P driven by the thinnest ultrasonic motor (0. 4mm thick). The weight of the small robot is 12.5g. It travels at a speed of 150mm/s. Due to a small size, light weight, and less energy consumption, aerial robots are not only suitable for space exploration, but also suitable for intelligence-gath-
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ering mISSIOn III military actions. Therefore. some countries are devoted to applying USMs to aerial robots. Fig. 15. 23 shows a Seiko Epson Corporation's uFR-II type flying robot driven by two ultra-thin USMs which drive two blades rotating in opposite directions to get lift. The largest diameter of the robot is 136mm. the height is about 85mm, total weight including battery is 8. 6g, and the flight time is about 3 minutes. A power-saving Bluetooth module is applied to multiple units to be remote-controlled. As a result of the ultra-thin USMs with the high performance. its lift is increased 30%[12:.
Fig. 15.22
Fig. 15.23
Monsieur II-P
Micro flying robot
A micro-moving robot made by Seiko Electronic Co., Ltd. is shown in Fig. 15. 24. The ultrasonic motor used for the robot is shown in Fig. 15. 25. A differential drive is adopted for this robot, where two driving whecls are used and each driving wheel is driven independently by one of the two USMs. The torque of the USM can be directly transferred to the wheel by teeth on the rotor of the USM. Due to its excellent maneuverability. the two motors are combined with each other and by gear coupling the rotors and wheels for output torque, the torque of the motor is converted to yield a linear motional force, and where only one wheel runs while the other is at rest then the micro-moving robot can turn round path. Because the micro-moving robot has given such excellent performance, it won first prize in a micro-robot competition in Japan- 3-.
Fig. 15.24
Micro-moving robot
Fig. 15. 25
USM used in micro moving robot
Figure 15. 26 shows a master-slave system driven by USMs developed by Keio
Ultrasonic Motors Technologies and Ap plicalions
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University of ]apan1l3 -. The five-fingered robot hand has totally 20 DOFs and eaeh DOF is driven by an USR30-B4 motor (maximum output torque about o. 1='J ·m). The mass of the entire hand is 853g. The width of the palm is 86mm, the length from fingertip to wrist is 203mm, and the length from the fingertip of thumb to the fingertip of the little finger is 190mm. Therefore the robot hand tha t has almost the same size of human hand is realized. Prior to this hand. according to the documents. the weight of the lightest hand driven by the traditional motor is 1 100g. Due to fast response of the USMs, the hand response speed is faster than a human hand. The positions of the master hand joints are obtained by trimmer potentiometers. A controller uses the position information to drive the dexterous hand. By means of force feedback from the master hand, human hand can feel the force of grasping. According to the grasping experience of human hand. further amendments can be transmitted to the joints of the master and slave hands, so effective and precise grasp or power grasp can be realized. This system is applied for remote master-slave control of equipments in medical instruments. nuelear industry. and space exploration.
Force feedback
Position feedback
Five-fingered robot hand(slave hand)
Fig. 15.26
15. 2. 4
Master hand
Master-slave system driven by USMs
Application in Surveillance Camera Platform
The surveillance camera platform is a device, which can be used for monitoring and real-time tracking a moving target, and it can fix and stall a camera. The platform's form can be divided into two types: fixed and rotating. The fixed platform is applied to small-scale scanning surveillance. and the rotating platform is applied to large-scale scanning surveillance and can realize real-time tracking a moving target. According to the feature of rotation, the rotating platform can be divided into horizontal rotation with I-DOF pan monitor platform and all-round rotation with 2-DOF pan tilt (PT) platform. Generally. the range of the rotary angle of the pan tilt monitor platform is from 0° to 350° and that of tilt angle is 90°. At present, the actuators for the pan tilt platform are two electromagnetic motors with speed reducers. For a scanning monitor platform with a constant rotation speed. the pan rate is usually from 3°/s to 10 /s and the tilt rate is around 0
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1 °/ s. For a real-time tracking platform with a variable rotation speed, the pan ratc is usually from OO/s to 32°/s and thc tilt ratc is from OO/s to 16°/s. In somc high-speed camera systems, the pan rate can be up to 180°/ s and tilt rate up to 120° / s. In a real-time tracking platform equipped with an electromagnetic motor, the variable speed control becomes difficult due to the motor's large rotational incrtia, and gear backlash can affect thc tracking effcct. If an elcctromagnetic stcp motor is chosen, the intermittcnt movemcnt of thc motor can causc thc image to vibrate, then information may bc lost. Compared with elcctromagnctic motors, USMs have good positioning characteristics and speed controllability, so they are more suited for the motion control of the camera platform. The camera platform driven by USMs possesses the following advantages-l1J : (1) USMs can dircctly drive the platform without spced-rcduction mechanism, so the whole structure could be simple and light weight. (2) The platforms driven by USMs gain higher precision and better stability. (3) USMs have a quick response, so they are more suitable for target tracking control. (4) USMs can not bc rotatcd by thc cffects of cxtcrnal environment (high humidity, vibration, ctc. ) under non-opcrating state, duc to thc self-locking without powcr. (5) Comparcd with thc electromagnetic step motors, the USM driving platforms can obtain morc completc imagc information. Figure 15. 27 shows the PT made by PDLab. It is driven by two ultrasonic motors and can rotate both in horizontal and vertical directions. It mainly consists of a camera and two ultrasonic motors (1 and 2). The rotation of the camcra in vertical dircction (tilt) is controlled by USMl, and horizontal rotation (pan) is controlled by USM2. Fig. 15.28 shows the block diagram of thc control systcm.
USM I
PC
CCD
Target
Fig. 15. 27 Surveillance camera platform
15. 2. 5
•
Pan-tilt
Fig. 15. 28 Block diagram of surveillance camera platform system
Applications in Precision Positioning Stage
Due to their characteristics of low speed and large torque without gear mechanism, and the screw-nut transmission can be omitted if the stage is drivcn by lin-
Ultrasonic Motors Technologies and Ap plicalions
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ear USMs. which can greatly improve the positioning accuracy of the stages. The stage can be widely used in biomedical, semiconductor manufacturing equipment, etc. The stages shown in Figs. 15. 29-15. 31 all are developed by PDLab. The stage shown in Fig. 15. 29 is driven by two rotary USMs. Its speed both in X and Y directions is 150mm/s and the positioning accuracy is 3fLm. The stage shown in Fig. 15. 30 is driven by two linear USMs. Its speed both in X and Y directions is 200mm/ s and the positioning accuracy is 2fLm. The X- Y-{) stage shown in Fig. 15.31 is driven by two linear USMs and one rotary USM. It can do two straight linear movements and a rotary one which is completed by the top disk of the stage.
Fig. 15. 29
Fig. 15.30
linear USMs
X-Y stagc using rotary USM and screw/nut
X- Y stage using two
Fig. 15. 31
X-Y-B stage using two linear USMs and one rotary USM
Unlike the structure of the XY positioning stage stacked up by two I-axis stages. ='Jational Taiwan University and Hcfei University of Technology of China have developed an XY positioning stage in one plane based on Abbe principle, as shown in Fig. 15.32. One mobile bar, supported by a precise linear skiing track, is driven by one linear ultrasonic motor. The platform is conducted in X, Y direction by the" push-pull" approach of mobile bars. The displacement of each axis is measured by nano-seale optical sensor which is installed at the symmetrical side of the platform. The transformation between X and Y movements is guided by precise linear tracks placed around the platform-15J.
Chapter 15
Applications of Ultrasonic Motors in Engineering
Fig. 15. 32
15. 3 15. 3. 1
461
Positioning stage in one plane
Applications in Biological and Medical Engineering Applications in Medical Facility
USMs neither produce a magnetic field nor subject to outside interference in the magnetic field, so it is very applicable to Magnetic Resonance Imaging (MRD. When patients are examined by MRI, they need the injection of a liquid in order to better observe the lesion. The speed of injection must be constant, so the best way is to adopt motors to injectors drive with a constant speed. Traditional electromagnetic motors have their own magnetic field and thus have a negative impact on real-time imaging. The previous methods inelude artificial injection and using a longer guiding rod between the electromagnetic motor and MRI. )Jow, USMs can be used for the injection. Fig. 15. 33 shows the apparatus for MRI injection. The ultrasonic motor (TRUM-60)J) used in the apparatus is made in PDLab. In order to ensure constant speed, control method is used to adjust the operating frequency of the motor in real time. Fig. 15. 34 shows the constant speed control performance of TRUM-60)J. It can be seen from the figure, at a certain range of torque, the speed of the ultrasonic motor can remain stable. USM
160 140 120
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100 ____ I _____ LI _____ I
__ _
€
80
0
60
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40
I I I'" I I I I I. ----~-----~-----~---I I I I
20
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"
0
~
0
~
I
I
I
I
I
..
I
I I
I I
I I
I I
0.3
0.6
0.9
1.2
---
1.5
Torque/(N'Ill)
Fig. 15.33 Injection device driven by TRUM-60N USM
Fig. 15.34 Constant speed control performance of TRUM-60N
In another application, the MRI-guided transurethral ultrasound therapy is a
Ultrasonic Motors Technologies and Ap plicalions
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minimally invasive treatment for localized prostate cancer is a now undergoing commercialization as Profound Medical Incorporation (PM!). This concept requircs the hardware that can operatc insidc thc bore of an MR imagcr during imaging with no mutual interference between the two systems. The prototype MRIcompatiblc rotary motor is built by using nonmagnetic components including HR2-N series :'\Ianomotion ultrasonic motors, which rotate a ceramic ring which provides thc rotational motion of the motor. Extremcly accuratc rotational vclocities havc bccn achieved a range from SO/min, up to 30r/min. The prccise control of thc rotational spccd, closing the position loop and thermal profile via thc MRI, allows hcalthcarc professionals to completcly destroy a tumor[l5: . Several prostate cancer treatments are available. including radiation. Even the treatments with high success levels leave a patient to suffer enduring and somtimes pcrmanent impotcncy and incontinence problems in the most cases. PMI's minimally-invasive thermal ablation device powered by :'\Ianomotion's motors treats prostate cancer as well as, and better than radiation. and projects to delivcr significantly fewcr sidc effects based on pre-clinical rescarch. While some radiation methods often require 9-12 weekly one-hour treatments, PMI's device complctes the trcatment proccss in just one visit, and with far greater accuracy for the targeting. Figure 15. 3S shows the picture of a 3-DOF ultrasonic motor. There is a spherical rotor at the top of bar shaped stator. It can rotate around .1':, y, and z axis. Fig. IS. 36 shows an example of application, in which the 3-DOF ultrasonic motor is used in a 7-DOF forceps for surgcry:s: Spherical rotor
Bar- haped stator
han
Fig. 15.35
15. 3. 2
Multi-DOF USM
Fig. 15.36 Multi-DOF forceps by using multi-DOF USM
Applications in Biomedical Engineering
The biomedical cngineering is onc of the hot research ficlds in recent years. Onc of its features is to rcquirc ultra-precision. for example, the precise driving stagc and precise feeding devices. USMs are paid extensive attention by the biomedical engineering rcsearchers in the world due to fast rcsponsc and high positioning accuracy. In gene transfering and artificial insemination process. the cell puncture is an indispcnsable micro operating. If the hydraulic actuator is uscd for thc cell opcrating systcm, thc wholc cell will have a largc deformation, as shown in
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Fig. 15.37. This large deformation makes the cell structure damage. Recently in the laboratory, Japanese scholars have developed an automated cell-puncture operating system which adopted nano-positioning technique based on the piezoelectric shock-type linear motor and image processing technique[17:, as shown m Fig. 15.38. The cell's deformation is greatly reduced in this operating system, as shown in Fig. 15. 39. The optical microscope is an essential equipment for pathology, cytology and metallography etc. In recent years, with the rapid development of computer image processing technology, the demand of the optical microscopes with computercontrolled precision motion in the x, y, and z directions is urgent.
Fig. 15. 37 Large deformation using conventional actuator
Fig. 15. 38 Cell puncturing system based on linear USM
Fig. 15. 39 Little deformation using linear USM
Comparing with the optical microscope driven by electromagnetic motors, the optical microscope driven by the USMs has a fast response in positioning, which can obtain much more clear image quality. Fig. 15. 10 shows an optical microscope driven by USMs made by PDLab.
Fig. 15. 40
Optical microscope driven by USMs
Due to the features of the small size, light weight and fast response, USMs have been favored in the rapid development of the imaging technique Optical
464
Ultrasonic Motors Technologies and Ap plicalions
Coherence Tomography (OCT). Compared with other imaging techniques, such as X-ray, CT, MRI, Ultrasound, etc. The OCT is promising a new noninvasive type of diagnostic medical imaging technology that utilizes advanced photonics and fiber-optics to obtain two or three dimensional images with extremely high, micrometer level resolution from living tissuse biological or human bodies. Tsinghua University of China is developing a prototype OCT endoscope, in which a micro ultrasonic motor is ineluded. The size of the motor is Imm in diameter, 5mm in length, and its weight is only 36mg. A schematic structure of the OCT endoscopic system is shown in Fig. 15. 11. 0 pl ical fiber
Transparenl coal
Fig. 15.41
AlIlo-zoom lens
Melal coal
USM
Micro mirror
Schematic structure of OCT system
Through the optical fiber, the infrared light is automatically focused on the auto-zoom lens and reflected to the observed target by a right-angle prism. Light scattered from the sample is selected by a confocal acquisition method, that is, the reflected light returns along the same path of the incident light. The rightangle prism is driven by the Imm diameter ultrasonic motor with a rotation speed around 360r/min. Through this design the optical tomography scanmng Image can be obtained from the circumference of the samples- 18J •
15. 4 15. 4. 1
Applications in Aerospace Engineering Applications in Aircraft
The flutter is a self-excite and potentially destructive vibration, which is very dangerous to air crafts. Once the flutter is generated, the aircraft will be crashed. How to prevent the flutter is a very important issue to be considered in aircraft design and testing. In addition to some traditional measures, one of the current advanced methods is to usc a control system that can put it down quickly at the beginning of the flutter, which is so-called active flutter suppression of aircraft. The active flutter suppression is a servo aeroelastic stability problem, which is related to the interaction between aircraft structure, aerodynamic force, and control system, including the combining action of elastic, inertial, aerodynamic, and control forces. Many specialist in the aviation have studied on the aircraft flutter suppression for many years. During the initial stage of research, hydraulic servo actuators arc used. With the constant emerging of new technologies and materi-
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als, various new types of actuators are developed. USMs become one of the best choices as actuator due to their characteristics of low speed and large torque. self-locking without power, fast response and high torque/weight ratio. Fig. 15. 12 shows the configuration of a two dimensional airfoil control system, in which a TRUM-45 made by PDLab is used in wind tunnel model test:]9]. Fig. 15. 43 shows the block diagram of the control system. Fig. 15.44 shows the base fixing airfoil and the detailed installation of the USM. Fig. 15.45 shows the driving and control device for the USM.
I
I
Control plate
Angle
I
I
PC
Instructio
""
Torque
Contro l plane
Angular displacement sensor
USM
Fig. 15.42 Configuration of two dimensional airfoil control system
Fig. 15.44 Conditions of fixing airfoil and installation of USM
I
USM
I I
Speed and dire{;tioll
USM control circuit
Fig. 15. 43 Block diagram of control system
Fig. 15.45 Driving and control cEvice for USM
DARP A/ AFRL/NASA Smart Wing program led by )Jorthrop Grumman Corporation (NGC) , aimed to develop novel and smart materials in order to control the surfaces that would improve aerodynamic. aeroelastic, and other system level benefits of military aircraft[20:. The program is divided into two phases. In phase 2, the main objective is to demonstrate high efficiency actuation of hingeless, spanwisc. and chordwise deformable control surfaces using smart materials-based actuators for a wind tunnel model with the 30 % of full scale aircraft proposed by NGC uninhabited combat air vehiele (UCA V) _2JJ. The model is shown in Fig. 15. 16(a). As shown in Fig. 15. 16(b), the smart materials based actuators are used for adaptively controlling the aileron in the right side of the model, while the conventional electromagnetic motors arc used for driving the aileron in the left side of the model. Smart material based confor-
466
Ultrasonic Motors Technologies and Ap plicalions
mal control surfaces based on ultrasonic motor adaptive control are integrated into the right side of the wing while conventional, hinged control surfaces driven by electric motors arc built into the left side, as shown in Fig. 15. 46(b). The R&D staff compared USMs with other smart materials-based actuators and conventional DC motors of similar size and power output. The results show that USMs have the power density higher than the smart materials based actuators and the electromagnetic motors. It should be noted that the high power density of USMs is due to the high frequency operating of the piezoelectric element. USMs are also competitive with the performance of many electromagnetic DC motors. USMs display a number of advantages over conventional motors and enable a simpler integration into the limited space available, such as in the wind tunnel model with 30 % full scale aircraft. The location of ultrasonic motors in aileron unit is shown in Fig. 15.47. The ultrasonic motor can accurately control bending and torsion of aileron. The single segment flexible structure of the model is shown in Fig. 15.18.
Conventional side
(a)
Fig. 15.46
Smart side
(b)
Wind tunnel model with 30 % of full scale aircraft
/
Fig. 15. 47 Actuation by distributed shear using USMs
15. 4. 2
Fig. 15. 48 Single segment flexible structure of model
Applications in Aerospace
Since spaeecrafts launch costs arc depended on primarily their mass placed on orbit. It is important that the mass of spacecrafts is as low as possible. Vacuum. along with a temperature range from +80'C to -150'C and high radiation level
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can wreak havoc on electromechanical motors that are used in the general circumstance. USMs have a lot of advantages. which reduce the spacecraft mass and the structural complexity, then improving reliability. In recent years, USMs have been applied in United States spacecrafts, such as Mars explorations[22:. Fig. 15. 19 shows the application of USMs in Micro lander for Mars Exploration jointly developed by Jet Propulsion Laboratory and the Massachusetts Institute of Technology. The torque of the ultrasonic motor is 2. 8)J'm, the usable temperature can be as low as - 100'C and its weight is lighter than conventional electromagnetic motors by 30 %.
Fig. 15. 49
Micro lander for Mars
NASA Coddar Space Flight Center of the United States has specially developed three ring-type traveling wave USMs, and their diameters are 1. 1 inch, 2. 5 inch and 2. 8 inch. respectively. They are used for a micro-manipulator on the space robot. as shown in Fig. 15. 50. The torque of these motors is 0.05N·m. o. ll)J'm, and o. 68N'm, respectively. The main reasons that )JASA adopted USMs as actuators for the joint of manipulator are as follows: first, the reliability of the USM in a vacuum is high; second, the stability of USMs is better 23J.
(a) Samplin g ann
Fig. 15. SO
(b) Three USMs lIsed i 11 robot
Micro lander for Mars and USMs used for the Mars
It has been reported that the USMs are more lighter by 10% than the traditional motors with the same function L23_ • With further development of USMs technology and the improvement of the motor, there is no doubt that USMs will be widely applied in many fields.
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References [ 1J
Chunsheng Zhao. Some proposals for development of ultrasonic motor techniques in China. Micromotors Servo Technique, 2006, 39(2): 64-67. (in Chinese)
[ 2J
I Okumura, H Mukohjima. A structure of ultrasonic motor for auto focus lenses. Motor Conference. Japan, 1987: 75-85.
[3 J [ 1J [ 5J [ 6J [ 7J [ 8J [9 J [lOJ
S Ueha, Y Tomikawa. Ultrasonic Motors: Theory and Applications. New York: Oxford Science Publications, 1993. I Okumura. A designing method of a bar-type ultrasonic motor for auto focus lcnse. The International Symposium on Theory of Machines and Mechanics. Nagoya, Japan, 1992: 836-841. T Maeno. Recent progress of ultrasonic motors in Japan. The 1st International Workshop on Ultrasonic Motors and Actuators. Yokohama, Japan, 2005: 15-17. Yinghui Dong, Chunsheng Zhao. Development of rocking head type ultrasonic motor at home and abroad. Small & Special Machines, 2003, 36(2): 35-37. (in Chinese) A lion, K Suzuki, M Kasuga, et al. Development of a self-oscillating ultrasonic micro-motor and its application to a watch. Ultrasonics, 2000, 38(1): 54-59. Hua Zhu, Zhihua Chen, Chunsheng Zhao. The application of ultrasonic motor in gasoline generator. Small & Special Machines, 2003(5): 25-26. (in Chinese) S Akio, M Mutsumi, M Hideyasu, et al. Steering wheel position adjusting apparatus for a vehicular steering system. United States Patent, 5806890, 1998-09-15. Hongjian Wang, Zhijun Sun, Chunsheng Zhao, et al. Movement control of autonomous mobile robot powered
[llJ [12J [l3J
[l1J
[l5J
by ultrasonic motors. ] uurnal of Vibration, Measurernent & Diagnusis,
2006,26 (3): 181-181. (in Chinese) Zhijun Sun, Rentao Xing, Chunsheng Zhao, et al. Fuzzy auto-tuning PID control of multiple joint robot driven by ultrasonic motors. Ultrasonics, 2007, 46(4): 303-312. Epson Develops World's Smallest Flying Mierobot. [2007-05-23J, http://ehina. nikkeibp. co. jp/ china/news/bps/bps200108200ll3. html I Yamano, T Maeno. Five-fingered robot hand using ultrasonic motors and elastic elements. Proceedings of the 2005 IEEE International Conference on Robotics and Automation. Barcelona, Spain, 2005: 2681-2689. Wenbo Gu. Target Identification and Tracking System Based on Ultrasonic Motors. Dissertation for the Degree of Master. Nanjing: Nanjing University of Aeronautics and Astronautics, 2007. (in Chinese) Guang M Fan, Fang Chen, Wei L Wang, et al. Key technology research of micro volume coordinate measuring machine. The Seventh Advanced Menu!acturing Technolugy Syrnpusiurn
of Main Land and Taiwan, 2008, 12: 21-21.
[l6J
[l7J [18J
[19J [20J [21J
J Medteeh's )Ion-magnetic Motors Power the World's First MRI-guided Prostate Cancer Treatment Device. [2009-10-12 J, http: / /www. johnsonmedteeh. com/news/press_jan21_ 2009. html T Higuchi. Automatic micro manipulation system for cell manipulation. [2007-05-23 J, http://www.intelleet.pe.u-tokyo.ac.jp/researeh/manipulator/manipulator_e. html Tieying Zhou, Kai Zhang, Yu Chen, et al. Development of Imm cylindrical ultrasonic motor and its application in OCT endoscopic. Chinese Science Bulletin, 2005, 50(7): 713-716. (in Chinese) Mingli Yu, Haiyan Hu. Active flutter suppression of an airfoil model using ultrasonic motor. Journal of Vibration Engineering, 2005, 18(1): 118-125. (in Chinese) J)I Kudva. Overview of the DARPA smart wing project. Journal of Intelligent Material Systems and Structures, 2004, 15: 261-269. D Jonathan, Bartley-Cho. Development of high-rate, adaptive trailing edge control surface for the smart wing phase 2 wind tunnel model. Journal of Intelligent Material Systems and
Chapter 15
Applications of Ultrasonic Motors
III
Engineering
469
Structures, 2004, 15:279-292.
[22J
[23J
Chunsheng Zhao. Development of mireo ultrasonic motor and its prospective applications in aerospace engineering. Symposium on Application of Micro-technology for Micro-sateliite. Beijing, 1997: 289-295. (in Chinese) S Sherrit. Smart material/actuator needs in extreme environments in space. Proceedings of the SPIE Smart Structures Conference. San Diego, CA, 2005: 5718-5761.
Index abrasion and fatigue, 15
autofocus system, 449
actuator, 2
automobile, 453
aging stability, 10
aerial robot, 157
anti-resonant frequency, 38, 361 anti-fatigue performance, 61
bar, 77
adhesive bonding techniques, 16
bar-type ultrasonic motor, 195
adhesion theory, 53
bar-type micro USM, 150
adhesi ve wear, 57
beam, 81
abrasive wear, 57
bending
anisotropy, 70
- vibration, 81
anisotropic tribomaterials, 71
- mode, 168,196
absorber, 112
bimorph actuator, 42
annular
bonding material, 167
- laminated plate, 133
boundary constraint, 208
- plate, 170
back-propagation neural network, 401
axial vibration amplitude, 177
booster, 377
APDL, 275
biomedical engineering, 462
AC voltage, 272 accumulated error, 301, 307
composite, 9
assembly and processing error, 337
- mode, 9
adjustable step length USM, 300
reverse piezoelectric effect, 1
acoustic
converse piezoelectric effect, 21, 265
- radiation pressure, 331,332
Curie-Weiss law, 25
- streaming, 331,332
corrosi ve, 58
- adhesive force, 331,333
coating, 60
- stream speed, 333
ceramics, 26
admittance, 34
circular plates, 86
- circle, 34, 35
ring plates, 88
- intensity, 439, 440
cylindrical shell's coordinate system, 93
adaptive controller, 386
cy lindrical shell, 92
average speed, 125
cylinder stator, 335
aerospace engineering, 464
characteristic matrix, 139
Index
471
circumferential speed amplitude, 162
- mIrror, 151
control system, 227
driver, 352
constrained variable metric method, 223
dry friction, 53
contact
distributed electric field, 96
- friction, 2
dynamic
- area, 211
- friction coefficient, 53, 51, 56, 70,
- region, 115
72, 73
- angle, 151
- characteristic, 111
- interface, 145, 150
- transmission, 131
- analysis, 215
disturbing mode, 272, 283
- model, 144, 243,292
interference mode, 169
- duration, 245
design variable, 271
- prcssure, 150
d'l effect, 270
compacting way, 308
d 33 effect, 281
elose loop, 392, 393, 105
differcntial composite motion, 307
complex programmable logic device, 358
differcntiation, 398
constant
disk stator, 337
- currcnt, 365
duty cyele, 357
- voltagc, 365, 370, 372
dead zone, 389
calcndar mechanism, 452
def uzzifica tion, 111
camera, 119
Doppler effect, 121
cell - phone, 151
electromagnetic
- puncturing system, 463
- actuator, 14
displacement
- principle, 13
- intcrfcrence, 1 - resolution, 8
elastic
- response, 283
- body, 1, 31
dielcctric
- coefficients, 70
- constants, 23
- modulus, 60
- loss, 23
energy
direct piezoelectric effect, 21
- transform, 13
domain, 26
- density, 2, 16, 61
driving
electrical quality factor, 23
- fcct, 265, 270, 272, 274,
electromechanical
281, 285 - mechanism, 312, 322
- coupling coefficients, 36 - coupling, 119
- torque, 331
elcctric
- tcsting, 338
- field intensity, 96
472
Ultrasonic Motors Technologies and Ap plicalions
- depolarization. 11
- response, 112, 172, 190
- displacement, 29
- difference, 271
equivalent circuit, 33, 363
finite element analysis, 210
- model, 117
forced vibration, 95, 335
elliptical
fixed step length USM, 301
- traj ectory, 111
feedback control, 393
- motion, 9, 12
frequency / voltage
elliptic motion, 232
transform a tion, 395
effective elliptical motion, 131
full bridge converter, 361
exei ting signals, 122
flowchart, 396
equilibrium position, 28
f uzzifica tion, 111
effective bending stiffness, 289
fuzzy
epoxy resin, 44
- controller, 387
eigenvalue, 185
- control rule, 412
extreme environmental conditions, 8
- decision, 415
environmental testing, 132
- domain, 112 - logic reasoning, 111
flexible sheet, 9
- subset, 112
flexible rotor, 165
fixed gain PID, 398
flexi bili ty coefficients, 28
five-fingered robot hand, 458
fatigue wear, 57
flutter suppression, 161
friction (frictional) - con tact force, 1
Guyan method, 110
- and wear behavior, 50
generalized coordinate, 141
- characteristics, 50
generalized eigenvalue problem, 206
- interface, 50, 308, 309
gap, 11
- drive, 8, 50
rayleigh wave, 313
- heat, 59
gasoline generator, 152
- noise, 59, 68
graded piezoelectric actuator, 42
- mechanism, 53 - coefficient regulator, 60
hi-tech, 1
- layer, 115
hydrothermal deposition, 11
- material, 65, 157, 167
hard assist materials, 58
- force distribution, 187
Hamilton principle, 132
- pair, 167, 259
Hermite polynomials, 135
- force, 50, 117, 189
half bridge converter, 361
finite length, 91
harmonic wave, 351
finite element method (FEM) , 76
hidden layer, 101
frequency
high/low temperature, 430
Index
473
half-sine wave, 137
- mode frequencies, 239, 211
headrest, 151
longi tudinal-torsional hybrid motor, 1, 10
hygrothermal environment, 110
longi tudinal- bcnding vibration, 9 in-planc
longi tudinal-torsional vibration, 10
- vibration, 12
L-T, 232
- natural vibration, 89
- hybrid vibration, 235
- mode, 92, 106
linear ultrasonic motor, 265
- bending modc shape, 91
- with a buttcrfly shapcd stator, 282
in tcrmi ttcn t opcra ting, 16
- with a whcel shapcd stator, 288
intcrfacc forcc, 146
lincar surfacc acoustic wavc motor, 340
inncr flangc structurc, 213
load charactcristics, 13, 261, 427
intcrpolation function matrix, 134
local stiffncss, 202
iterative process, 209
LE type, 270
interlaced and connected electrodes, 311
lamb wave, 313
IDT, 311
Lagrange function, 132
impcdancc, 33
Lyapunov function, 409
intcgration cocfficicnt, 398
lifc tcsting, 444
industrial cnginccring, 452 isotropic tri boma tcrials, 71
morphotropic phase boundary, 22
jacketed stator, 280, 290
metal
kinctic cncrgy, 97
- core piezoelectric fiber, 11
- film, 3 Kirchhoff Assumption, 83
mcchanical
lcad zircona tc-lcad ti tana tc, 22
- quality factor, 22
lead-free piezoelectric materials, 15
- characteristics, 58
layer, 13
- depolarization. 11
longitudinal
mode (modal)
- vibration, 2, 31, 221
- shapcs, 78
- modc, 9, 196
- scparation, 174
- (piezoelectric) effect, 32-33
- force, 98
- strain mode, 101
- convertor, 255
- wave, 109, 256
- parameters, 78
- ccramic picccs, 239
- mass, 80
- (vibration) amplitudc, 247, 252
- stiffncss, 80
- extension condition, 235
- analysis, 76
- displacement, 237, 213
- testing, 338
longitudinal and torsional
- rotary, 321
- vibration, 232
- ncphogram, 179
474
Ultrasonic Motors Technologies and Ap plicalions
- inferences. 202
- of elastic body. 75
- frequency modification. 210
- of plates. 82
- characteristics. 80
- of rectangular plates. 83
- frequency. 142
- of circular plates. 86
mode conversion
- of circular ring plates. 88
- type. 232. 255. 258
- of thin plates. 89
- type ultrasonic motor. 255
- of cylindrical shells. 92
- LTUM. 258 mass. 97
node (nodal)
motion
- circle. 121
- mechanism. 121
- diameter. 121
- trajectory. 117. 123
non-centrosymmetric point group. 25
- locus. 387
non-polarization. 129
mechanism. 76
different amplitudes. 125
multi-mode type. 232. 235. 239
non-contact. 330
- LTUM. 235
- driving model. 333
multi-objective optimization. 203
- type ultrasonic motor. 327
multi-con tact-point slider. 348
nonlinear
multivibrator. 353
- control, 387
MAC. 207
- spring. 261
mover. 272
notched
medium of water. 312
- cylinder. 288
matching circuit. 351
- stator. 289
mixed controller. 387
nano-meter step USM. 300
membership function. 412
no-load speed. 156. 247.250
model reference adapting control. 105
neutral layer. 121
medical facility. 161
neural network controller. 386
moving robot. 157
numerical simulation. 126
manipulator. 18
nOIse
monsieur II-P. 457
- measurement. 438
measurement methods. 111
- test. 138
moisture absorption -characteristics. 110
operating frequency. 2
- ratio. 441
operating mode. 167 operator matrix. 130
natural
orthogonali ty of mode shapes. 78
- modes. 75
orthogonal
- frequencies. 75
- modes. 85. 121. 130
natural vibration. 75
- bending modes. 122
Index
475
out-of-plane. 82
piezomotor. 2
- vibration. 82
PZT pieces. 99
- bending vibration. 83
plane and spherical waves. 108
- bending mode shape. 90
poling process. 22
- bending mode. 171
polarization pattern. 128
output
pertur ba tion. 173
- efficiency. 66
potential energy. 96. 137
- torque. 218
phase difference. 272. 281
optimal
- adjusting. 389
- design. 176
preload. 63
- variables. 207
pre-pressure. 269
- model. 209
performance
- algorithm. 208
- simulation. 118. 214
objective function. 208
- measurement. 225. 336
outer flange structure. 213
perpendicular bending modes. 197
oblique ellipse. 126
pattern search algorithm. 208
open-loop control, 301
position
optical microscope. 463
- resolution. 301
OCT endoscope. 464
- control, 390 - error. 301
phys-chemical property. 51
- adjusting apparatus. 455
polyvinylidene fluoride. 22
posi tioning accuracy. 7
polymer. 60
precision positioning. 301
polytetrafluoroethylene. 61
parallel
plasma spray process. 65
- resonance frequency. 31
parametric analysis. 118
- capacitance. 139
piezoelectric
- capacitor. 370
- actuators. 2
- inductor. 372
- elements. 39
power amplifier. 371
- vibrator. 2. 21. 42
power transmission. 115. 200
- composite stator. 3
- efficiency. 145
- equations. 28
PID controller. 386. 400
- stack actuator. 11
positive definite matrix. 410
- fiber actuator. 43
proportion coefficient. 398. 102
- composite. 46
push-pull converter. 360
- material, 11. 21
phase frequency curve. 368. 372
piezoelectric ceramic - cylinder. 195
quartz. 21
- ring. 288
quasi-static friction. 72
476
Ultrasonic Motors Technologies and Ap plicalions
rochelle salt. 21
- wave. 109
ring-type. 5
- acoustic wave. 17. 327. 310
- USM. 119
shear
rotary
- wave. 109
- tribometer. 63
- vibration. 239
- SAW USM. 311
superposition of waves. llO
relaxor ferroelectrics. 22
static
running life. 58
- friction. 53
reinforced filler. 60
- friction coefficient. 51. 69
rectangular plates. 83
quasi-static friction. 72
rigid rotor. 145. 152. 183
stick-slip. ll8
resonant frequency. 362
sign function. 128. 118
resonance
semi-analytical
- frequency. 33
- annular element. 133
- voltage step-up. 378
- electromechanical coupling model, 132
- cycles. 379
substructural modal synthesis method. 133
response time. 397. 416. 424
substructure. 133
radial
- interface loading theory. 137
- friction. 145. 151, 165. 184. 186
structure (structural)
- motion. 200
- parameter. 166. 171
- relative slip. 213
- design. 239
random
structural perturbation theory. 180
- excitation. 135
sensitivity analysis. 172. 207. 275. 286
rayleigh wave. 341 rectangular thin plate. 270
stiffness matrix. 28.137
robot arm. 395
sequential quadratic programming. 176
rearvlew mIrror. 151
startup
stress-free dielectric constant. 21
- response.217. 121
short-eireui t
- time. 303. 307
- compliance. 29
- stage. 302. 304
- stiffness. 29
- response time. 121
standing wave. 3
shutdown
- USM. 300. 309
- time. 291
shafts. 80
- stage. 302. 305
sliding
- response time. 424
- surface. 51
strain mode. 80. 105. 271
- friction. 53.302
- shape. 271
surface
symmetry mode. 282
- roughness. 70
anti-symmetry mode. 282
Index support plate, 290
477
transient
synergetic operating technique, 296
- response characteristics, 14
single resonant excitation, 297
- response, 154
single non-resonant excitation, 298
- characteristics, 119
sweep excitation, 297
transverse vibration, 31
indi vid ual exci ta tion, 297
transformation matrix, 136
step
transfer function, 408
- control, 301, 303
tribology, 50
- ultrasonic motor, 300
tribomaterial, 50
steady
tribosystem, 50
- operating stage, 302
tri bopair, 52
single-step, 301, 305 - operating error, 301 - positioning accuracy, 308 signal generator, 305, 352 stability - of stator's vibration, 308 - of friction interface, 309 self-correction - nodal-line type step USM, 325 - peak step USM, 325 - USM, 300 self-tuning control, 110 solid to - fluid to solid way, 327 - solid way, 327 senes - capacitor, 368, 373
thin films, 11 thin plate, 83 three dimensional motion, 131 torsional - vibration, 76, 80, 232, 235, 211 - velocity, 235 - ceramIc pIeces, 239 - vibration amplitude, 219 thermal - depolarization, 10 - stability, 61 - cyeling experiments, 419 traveling waves, 107 - rotary ultrasonic motor, 88 tangential - speed, 132 - velocity, 200 - and normal direction, 293
- inductor, 377
- vibration velocity, 245
stall torque, 50
trajectory tracking, 227
slider, 266, 289, 302
triangle structure, 278
spherical rotor, 219
thrust-weight ratio, 281, 292
state of the art, 310
vibration type, 281
shift register, 356
thrust in steady state, 292
servo control, 386
teeth's
tuning fork watch, 2
- translational movement, 310
tra veling-wa ve ultrasonic motor, 5
testing techniques, 419
- rotary movement, 311
torque density, 7
478
Ultrasonic Motors Technologies and Ap plicalions
uniform thin plates, 83
virtual
ultrasonic range, 120
- mechanical and electrical work, 205 - work principle, 216
vibration
variational work, 132
- type, 9
variable gain PID, 100
- equations, 83
vacuum environment, 119, 132
- response, 76
violent vibration, 430
- amplitude, 2, 162, 167, 370, 387 - environment, 434
wear
- grades, 135
- mechanism, 56
- level, 435
- resistance, 50
- alarm, 151
wave
- motor, 152
- propagation, 107
vibrating speed, 333
- theory, 107
vibrator, 3, 196
- in elastic body, 108
voltage - controlled oscillator, 355
X-y stage, 160
- amplitude adjusting, 387
X-Y-8 stage, 160
Appendix A
Natural Vibration Frequencies and Mode Shape Functions of Bars, Shafts, Beams and Plates Table A. 1 Longitudinal vibration natural frequencies and displacement mode shape functions of a uniform bar with three boundary conditions Boundary conditions
"fatural frequencies _ C2n-lh 21
Wn -
(E
Y
p
Displacement mode shape functions
~n(.T) ~ sinC2n-l) ~7.n ~ 1.2.3.···
ITX
.I.
'rnCr) =cosnT,n= 1,2,3,'"
Table A. 2 Longitudinal vibration natural frequencies and their strain mode 1unctions 01 a uniform bar with three boundary conditions Boundary conditions
='Jatural frequencies
Strain mode shape functions
C2n-l)IT 21 eosC2n-l)
tn(.T) ~
-n-rr.
7tX -I-smn/.n
IT.T Zz.n
~
~ 1.2.3···
1.2.3.···
Table A. 3 Torsional vibration natural frequencies and their mode displacement shape 1unctions 01 a circular and uniform shaft with three boundary conditions Boundary conditions
"fat ural frequencies _ W,,-
_ W,,-
nIT
rc; Yp
nIT
rc; Yp
I
I
_ C2n-lh 21
Wn -
.I. C ) 'rn.T
.I. 'fu(.L)
rc;
Y
Displacement mode shape functions
p
ITX =cosnT,n= 1,2,3,'"
. ITX 0 = slnnT,n = 0 , 1 , 2 ,~),'"
~,,(.T) ~ sinC2n-l) ~·7.n ~ 1.2.3.···
Ultrasonic Motors Technologies and Ap plicalions
480
Table A. 4
Values of first five X~ and characteristic equations of bending vibration
of a uniform beam with common boundary conditions Boundary conditions
Characteristic equations
xi
xl
x;
x;
X~
+ coshXeosX ~
3. 516
22.03
61. 69
120.9
199.1
i
1
J;
1
i
J
Table A.5
sinX
0
0
~
(rr2 )
(nrr) 2
(3rr2 )
(1rr)2
(5rr) 2
9.869
39.17
88.82
157.9
216.7
1 - coshXeosX
~
0
22.37
61. 67
120.9
199.8
298.5
1 - CDshX eosX
~
0
15. 41
49.96
104.2
178.2
272.0
Displacement mode shape functions of bending vibration of a uniform beam
with common boundary conditions Displacement mode shape functions
Boundary conditions .T .T
¢,,(.T)
~ 0, ¢(O) ~ nO) ~ 0 ~
L, ¢f'(L)
~ ¢QI(L) ~
0
Fn[eoshfln.T - eosfl"or - en (sinhflnx - sinflnx) ]
G ~ sinhflnL - sinfl"L
n
1
~
eoshflnL
+ e()Sfln L
x ~ 0, ¢(O) ~ ¢f'co) ~ 0
.T
~
L, ¢(L)
.T .T
~
0, ¢(O) ~ ¢/(O) ~ 0
~ ¢f'(L) ~ 0
~ L, ¢(L) ~ ¢/(L) ~ 0
x ~ 0, ¢f'CO) ~ ¢f'CO) ~ 0
.T
~
L, ¢f'(L)
~ ¢QI(L) ~
0
Fn[eoshfln.T - eosfl"or - en (sinhflnx - sinflnx) ]
¢,,(.T)
~
C Tn -
eoshflnL - e()Sfl"L
¢n(X)
~
sinhflnL - sinflnx Fn[eoshflnx - G" (sinfl".T
en ~ sinhflnL
+ eosflnx + sinhfln.T) ]
+ sinflnL
eoshflnL - e()sfl"L
Table A. 6
Constants A= for first five natural vibration frequencies of a circular plate (1) Fixed edge 0
o
2
4
2
4
3. 196
4.611
5.906
7. 144
8.347
9.526
6.306
7.799
9. 197
10.537
11.837
13. 107
9.319
10.958
12.102
13.795
15.150
16.175
12.577
14. 109
15.580
17.005
18. 396
19.759
15.716
17.256
18.744
Appendix A
Natural Vibration Frequencies and Mode Shape ...
Table A. 6 (continued)
481
(2) Hinged edge
~
0
1
2
3
1
5
0
2.222
3.728
5.016
6.321
7.539
8. 729
1
5. 452
6.963
8.374
9.724
11. 032
12. 309
2
8.611
10.318
11. 589
12.987
11.318
15.677
3
11. 761
13.297
11.772
16.201
17.596
18. 961
4
14.907
16.449
17. 940
19.391
5
18.051
19.598 (3) Free edge
~
2
0
o 2
1
Table A. 7
4
2.315
3.527
4.673
5. 788
3.001
1.525
5.938
7.281
8.576
9.836
6.200
7.731
9.185
10.580
11. 931
13.257
9.368
10.907
12.382
13.809
12.523
11.067
15.558
17.006
:"I atural vibration frequency constants Amn of out-of-plane vibrations of a
thin and circular ring plate with fixed internal and free external circumferences b/a
~
1
2
3
1
5
0.1
0 1 2
1. 865 5. 261 8.803
2.371 6.078 9.497
3.529 7.294 10.632
1.673 8.577 11. 940
5. 788 9.837 13.257
0.2
0 1 2
2. 194 5.876 9.817
2.540 6.478 10.302
3.552 7.444 11.073
4.675 8.614 12.110
5. 788 9.844 13.306
0.3
0 1 2
2.560 6. 681 11.217
2.821 7.138 11. 536
3.644 7.877 12.070
4.699 3.839 12.809
5. 793 9.939 13.726
Ultrasonic Motors Technologies and Ap plicalions
482
Table A. 8
Natural frequency constant Am" of in-plane vibrations of a thin and circular
ring plate with both free conditions at internal and external circumferences
~
All
b/a
0 0.1 0.2 0.3 0.1 0.5 0.6 0.7 0.8 0.9
~ b/a
0 0.1 0.2 0.3 0.1 0.5 0.6 0.7 0.8 0.9
1. 1. 1. 1. 1. 1. 1. 1. 1. 1.
A21
602 37 612 50 636 78 661 46 670 00 651 37 609 62 546 02 47781 108 10
3.523 3.636 3. 838 3.899 3.999 4. 322 5.021 6. 388 9. :,2:,
A32
1. 183 4.067 3.994 1.371 5. 018
A31
71 98 40 73 18 60 62 87 14
1.000 57 4. 027 42 4.15371 1. 580 51 5. 280 05 6. 303 95 7.862 98
A13
89 21 37 23 21
5. 154 45
5.610 82 6.775 18 9.550 11
2.101 2.097 2. 043 1. 867 1. 601 1. 315 1. 033 0.751
A 12
1. 367 93 1. 287 45 1. 097 33 0.895 19 0.715 09 0.55671 0.41642 0.295 63 O. 209 30 0.17118 A23
12 24 17 07 21 98 30 76
3.11981 3.415 76 3.366 15 3.291 31 3. 309 05 3.367 90 3.102 35 3.370 76 3.275 07 3. 142 81
A22
2. 189 12 2.43701 2.378 10 2.387 07 2.126 71 2. 458 93 2. 151 62 2.105 12 2.322 67 2.223 56 A33
5. 295 5. 222 1.732 1.619 5.088
27 25 08 19 74
5.965 45
6.120 91 7.35023 9.910 19
Appendix B
Natural Vibration Mode Shapes of Bars, Shafts, and Beams
Fig. B. 1 First 10ur displacement mode shapes 01 longitudinal vibration 01 a uniform bar with three boundary conditions
Fig. B. 2 First 10ur strain mode shapes 01 longitudinal vibration of a uniform bar with three boundary conditions
Ultrasonic Motors Technologies and Ap plicalions
484
--......-:Si
~SO
~ 13 O'~.,!!;r
SO
~\ 0.36
0.91
~09 V{j64
V
~V
0.28 0.72 ~071Ai50 JAi93
0.36 0.64 0.91
-""""I
crv,
~
~\
0.50 28
d
0.93
1Ai72 V
~V
0.41
0.77
A 1Ai b. 23
59
V\.J
0.40
0.80
d Idi 10 20
60
VV
0.23
0.59 0.95
\061Ai 41 1A77 v VV V
Fig. B. 3 First five displacement mode shapes of bending vibration of a uniform beam with common boundary conditions
-0.22
~SO
" "
0.36
0.13
050
...........Ie:---.
~09
v
0.91
lOr V v
0.36 0.64
~ 0.50
0.09 0.36 0.64 \! !.-.J
""'"""""'" 0.28
0.07
0.72
0.50
\!V "" V""'"'-
0.28
0.23
0.59 0.95
V
\TV
0.41
!/""\. \J 0.50
0.72
vII'"'\!vIC\ V61&411&77v
0.36 0.64
.t"")J
d V28 1diV72
0.77
0Jo\ O,fAs vvv
0.41
0.77
A23~59b.
\TV
Fig. B. 4 First five strain mode shapes of bending vibration of a uniform beam with common boundary conditions
Appendix C
Natural Vibration Displacement and Strain Mode Shapes of Plates, and Their Nephogram
l\o(8 101 Hz)
B"
(16 753 Hz)
B.,(18 719 Hz)
BIZ (21 070 Hz)
B'o(24 728 Hz)
B,,(3O 570 Hz)
B22(34 705 Hz) (a) Displacemet modes
Fig.
c. 1
(b) Displ. mode nephograms
(c) Displ. node patterns
(d) Strain mode nephograms
First eight displacement and strain modes of the out-of-plane vibration
of a thin and rectangular plate with free boundary condition on all sides
486
Ultrasonic Motors Technologies and Ap plicalions
BII (10 971 Hz)
Bx(16 31S Hz)
Bo (20 60S Hz)
B,,(27 798 Hz)
an (48 929 Hz)
B... (49 167 Hz)
B.(49 167 Hz)
B110(6O 383 Hz)
(a) Displacemet modes
(b) Displ. mode nephograms
(e) Displ. node patterns
(d) Strain mode nepbograms
Fig. c. 2 First eight displacement and strain modes of the out-of-plane vibration of a thin and square plate with free boundary condition on all sides
Appendix C
Natural Vibration Displacement and Strain Mode ...
487
Bo.(OHz)
Bm{3 993 Hz)
Boo(9 093 Hz)
Soc(IS 604 Hz)
Bns{23 349 Hz)
80.(32 ISS Hz)
807(41895 Hz)
Bos(52 441 Hz) (a) Displacemet modes
(b) Displ. mode nephograms
(e) Displ. node patterns
(d) Strain mode nephograms
Fig. C. 3 First 16 displacement and strain modes of the out-of-plane vibration of a thin, circular, and solid plate with free circumference condition and with nodal diameters only
488
Ultrasonic Motors Technologies and Ap plicalions
© 0 E9
811)(6864Hz)
8,,(15000 Hz)
Bli24 758 Hz)
@
Bw(27 055 Hz)
®
BIl(35 687 Hz)
@
811(40 132 Hz)
®
B14(47 465 Hz)
8n(53 821 Hz)
(a) Oispiaoemet modes
(b) Oispl. mode nephograms
~
(c) Oispl. node patterns
(d) Strain mode nephograms
First 16 displacement and strain modes 01 the out-oI-plane vibration 01 a thin, circular, and solid plate with free circumIerence condition, and with nodal diameters and nodal circles
Fig. C. 4
Appendix C
:"Iatural Vibration Displacement and Strain Mode ...
489
gOgO 20©0 9.0©0 o O©O o 000 o 000 o 000 gooo 800(15 869 Hz)
1304(20 193 Hz)
8",(26 749 Hz)
8a6(3S 313 Hz)
(a) Oisplacemet modes
(b) Oispl. mode nephograms
(c) Oispl. node patterns
(d) Strain mode nephograms
Fig. C. 5 First 16 displacement and strain modes of the out-of-plane vibration of a thin and circular ring plate with the fixed internal circumference and free external circumference conditions, and with nodal diameters only
490
Ultrasonic Motors Technologies and Ap plicalions
9.0@0 9.0@0 ~O@O
9.0@0 9.000 9.000 qooo qooo
(a> Oisplacemet modes
(b) Oispl. mode nepbograms
(e) Oispl. node patterns
(d) Strain mode nephograms
Fig. C. 6 First 16 displacement and strain modes of the out-of-plane vibration of a thin and circular ring plate with the fixed internal circumference and free external circumference conditions, and with both nodal diameters and nodal circles
Appendix C
Natural Vibration Displacement and Strain Mode ...
0 (Ql 0 00 f©~ 0
491
~ s.,(23 134 Hz)
_~
~o@o
qO©O oO@O 8,.(71194 Hz)
~o@o ~o@o ~o@o
(a) Di8p1acemeut IIIOIIes
(b) Di8pl. mode nephosJ:am8
(c) Displ. node paIIem8
(d) SInlio mode nephosJ:am8
Fig. C. 7 First 8 bending displacement and strain modes of the in-plane vibration of a thin and circular ring plate with the free-free circumference conditions
Ultrasonic Motors Technologies and Ap plicalions
492
o o o o o o o o
E.(2S 026 Hz)
E..(32 714 Hz)
E",(49693 Hz)
E..(68 796 Hz)
T,.(92 021 Hz)
TII(97265 Hz)
T,,(110 793 Hz)
T,,(129 098 Hz)
o 0 o@o o©o o@o 00 000 000 000
(b) Diapl. mode~
Fig. C. 8
(e) Diapl. nodepauems
(d) Straillmodenephograms
First 10ur extention-contraction (Eoo -Eo3 ) and torsion ('1'10-'1'13) displacement and
strain modes 01 the in-plane vibration 01 a thin and circular ring plate with the free-free circumference conditions
Appendix C
Natural Vibration Displacement and Strain Mode ...
493
QO@O QO@O o o@o 9.0©O B..(SO 147 Hz)
QO@O
QO@O o o 0 o o@o E".(73 412 Hz)
E,y.(J4 330 Hz) (a) Displacemet modes
(b) Disp\. mode nepbograms
(e) Displ. node patterns
(d) Strain mode nephograms
Fig. C. 9 First 8 displacement and strain modes 01 the in-plane vibration 01 a thin and circular ring plate with the fixed internal circumference and the free external circumference conditions
494
Ultrasonic Motors Technologies and Ap plicalions
EB
@)
o
@ ~ '<1Y
~ 'eT5;j
B,.(48 741 Hz) (a) Disp1acemet modes
@ (b) Displ. mode nephograms
(e) Displ. node patterns
(d) Strain mode nepbograms
Fig. C. 10 First 8 bending displacement and strain modes of the in-plane vibration of a thin, circular, and solid plate with a free external circumference condition
